High Frequency Electron Dynamics in Thin Film Superconductors and Applications to Fast, Sensitive THz Detectors
A Dissertation Presented to the Faculty of the Graduate School of Yale University in Candidacy for the Degree of Doctor of Philosophy
By Peter John Burke Dissertation Director: Professor Daniel E. Prober December, 1997
c
1998 by Peter John Burke All Rights Reserved.
Acknowledgments
This thesis work is the result of many individual e orts combined. I must rst thank my collaborators at JPL, namely Rob McGrath, Anders Skalare, Boris Karasik, Bruce Bumble, and Rick LeDuc. My collaboration with them was very productive and enjoyable. During my visits to JPL, many conference calls, and meetings, they were always willing to help support my thesis research in any way they could, and they did. Additionally, I have enjoyed the company and support of the entire cast of characters in Becton Center over the years. I was very impressed by the willingness of everyone I worked with at Yale to take time out from their busy schedules to help x this or that piece of equipment or understand this or that theoretical concept. I have tried to emulate that behavior myself, and reciprocate some of the assistance I have been given. I must also provide special thanks to Jean Belfonti for guiding me through the Yale bureaucracy. I have also enjoyed immensely the people I worked closely with in ProberLab, namely Lianne Verheijen, Alex Kozhevnikov, and especially Rob Schoelkopf. The rst thing Rob taught me was how to make SMA connectors, but that was just the beginning. I have continued learning from him about topics mundane and profound alike. I hope that will continue. The relationship between advisor and student is in many ways like the relationship between parent and child. A child receives more support and care from the parent than it can repay, and when mature it leaves the parent with only memories. My relationship with Dan Prober was no exception. He sel
essly provided me with more support and training than I can ever hope to repay, to which I can only say: Thanks. I am also happy to acknowledge the State of Connecticut and NASA for nancial support. iii
iv
Finally, my parents have provided me with an enormous amount of support, both moral and nancial, and have always merrily put up with my complaints about graduate school, providing advice and reminiscing about their own experiences as graduate students when necessary. I can only imagine how dicult it was for them to go through graduate school with me screaming in the back seat. They instilled in me the work ethic required to complete this thesis, and so in many ways it is not only my work, but theirs also. I hope they are proud.
Contents
Acknowledgments List of gures List of tables List of symbols and abbreviations 1 Introduction
1.1 Motivations and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 1.1.2 1.2 Emission in the Sub-mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Detection in the Sub-mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii xii xiii xiv 1
2 2 3 8 8 9 9 10 11 13
Existing technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 1.2.2 SIS tunnel junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schottky diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3
Hot-electron bolometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 1.3.2 Semiconductor based HEBs: InSb, AlGaAs . . . . . . . . . . . . . . . . . . . Superconductor based HEBs: Nb, NbN . . . . . . . . . . . . . . . . . . . . .
1.4
Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Theory
2.1 Mixing in bolometers: generic description . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Calculation of conversion eciency . . . . . . . . . . . . . . . . . . . . . . . . v
14
14 15
vi
2.1.2 2.1.3 2.1.4 2.2
Thermodynamic temperature
uctuations: Langevin approach . . . . . . . . E ects of electro-thermal feedback on output noise . . . . . . . . . . . . . . . Mixer noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20 24 25 26 27 27 28 28 31 31 34 36 39 40 40 42 44
Mixing in superconducting hot-electron bolometers: electron and phonon dynamics . 2.2.1 2.2.2 2.2.3 2.2.4 Electron-electron interaction rate . . . . . . . . . . . . . . . . . . . . . . . . . Electron-phonon interaction rate . . . . . . . . . . . . . . . . . . . . . . . . . Electron-phonon and electron-electron interaction lengths . . . . . . . . . . . Thermal time constant: di usion and phonon cooling . . . . . . . . . . . . .
2.3
Temperature Pro le . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5 Strong and weak dc heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weak ac heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spatially distributed temperature
uctuations . . . . . . . . . . . . . . . . . . DC heating with electron-phonon interaction . . . . . . . . . . . . . . . . . . AC heating in the presence of electron phonon interaction; strong AC heating
2.4 2.5 2.6
Uniform vs. non-uniform dissipation of power . . . . . . . . . . . . . . . . . . . . . . DC and ac di erential impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Voltage dependence of gain and noise;
from I-V curve
. . . . . . . . . . . . . . . .
3 Experimental technique
3.1 RF setup 3.1.1 3.1.2 3.2 3.3 3.4 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . RF block diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mixer mount . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
47 47 50 54 56 56 57 58 58 63
DC electronics
Cryogenic setup and thermometry
Shielding and ltering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 3.5.2 3.5.3 Calibrations using a coherent signal . . . . . . . . . . . . . . . . . . . . . . .
Calibrations using incoherent signals . . . . . . . . . . . . . . . . . . . . . . . Drift of calibrations during run . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
4 Device geometries and dc properties
4.1 4.2 4.3 Fabrication and geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resistance vs. Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Current-Voltage Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
66 68 69
5 Measurement of gain, noise, and bandwidth
5.1 Gain and noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 5.1.2 5.1.3 5.1.4 5.1.5 5.2 5.3 5.4 I-V curves vs. bath temperature, LO power . . . . . . . . . . . . . . . . . . . Gain and noise vs. voltage, LO power . . . . . . . . . . . . . . . . . . . . . . Gain vs. intermediate frequency . . . . . . . . . . . . . . . . . . . . . . . . .
72
73 73 78 93 94 107 109 116 120
Noise vs. intermediate frequency . . . . . . . . . . . . . . . . . . . . . . . . . Noise vs. LO frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Device impedance measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Normal state noise thermometry measurements . . . . . . . . . . . . . . . . . . . . . Comparison with theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Comparison to JPL THz Receiver Measurements 7 Conclusions
7.1 7.2 Summary of results presented in this thesis . . . . . . . . . . . . . . . . . . . . . . . Suggestions for future experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . .
137 142
142 145
A Calculated values of and 0 from I-V curve. Bibliography
147 159
List of Figures
1.1 1.2 2.1 2.2 2.3 2.4 2.5 2.6 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 Schematic emission in the Sub-mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic of heterodyne detection system . . . . . . . . . . . . . . . . . . . . . . . . Mixing in bolometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal circuit diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature pro le without electron-phonon interaction . . . . . . . . . . . . . . . . Distributed vs. lumped element temperature rise . . . . . . . . . . . . . . . . . . . . Temperature pro le with electron-phonon interaction . . . . . . . . . . . . . . . . . . Equivalent circuit of bolometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Block diagram of RF measurement system . . . . . . . . . . . . . . . . . . . . . . . . Mixer mount, side view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 5 16 29 33 37 41 43 49 51 52 53 55 59 60 62 64 65 68 70
Mixer mount, top view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lead geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DC electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Calibration coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calibration at xed frequency of ampli er gain and noise. . . . . . . . . . . . . . . . Calibration of ampli er gain using coherent vs. incoherent source. . . . . . . . . . .
Electron temperature vs. applied power, dc and ac. . . . . . . . . . . . . . . . . . . .
3.10 Drift of ampli er gain and noise with time . . . . . . . . . . . . . . . . . . . . . . . . 4.1 4.2 SEM of a Nb microbridge with gold pads. . . . . . . . . . . . . . . . . . . . . . . . . Resistance vs. temperature curves for di usion-cooled devices. viii . . . . . . . . . . . .
ix
4.3 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9
I-V curve of device A1 for bath temperature of 2 K. . . . . . . . . . . . . . . . . . . I-V curves at di erent bath temperatures for device A1. . . . . . . . . . . . . . . . . I-V curves under di erent LO powers for device A1. . . . . . . . . . . . . . . . . . .
71 76 77 82 82 83 83 84 84 85 85 86 86 87 87 88 88 89 89 90 90 91 91 92 92
Eciency, output and mixer noise for device A1 vs. LO power. . . . . . . . . . . . . Saturation curve for device B. DC bias point = 0.5 mV. . . . . . . . . . . . . . . . . Gain and noise vs. voltage for device A1 in overpumped case. . . . . . . . . . . . . . Current vs. voltage for device A1 in overpumped case. . . . . . . . . . . . . . . . . . Gain and noise vs. voltage for device A1 in optimum gain case. . . . . . . . . . . . . Current vs. voltage for device A1 in optimum gain case. . . . . . . . . . . . . . . . . Gain and noise vs. voltage for device B in overpumped case. . . . . . . . . . . . . . .
5.10 Current vs. voltage for device B in overpumped case. . . . . . . . . . . . . . . . . . . 5.11 Gain and noise vs. voltage for device B in optimum gain case. . . . . . . . . . . . . . 5.12 Current vs. voltage for device B in optimum gain case. . . . . . . . . . . . . . . . . . 5.13 Gain and noise vs. voltage for device C in overpumped case. . . . . . . . . . . . . . . 5.14 Current vs. voltage for device C in overpumped case. . . . . . . . . . . . . . . . . . . 5.15 Gain and noise vs. voltage for device C in optimum gain case. . . . . . . . . . . . . . 5.16 Current vs. voltage for device C in optimum gain case. . . . . . . . . . . . . . . . . . 5.17 Gain and noise vs. voltage for device D in overpumped case. . . . . . . . . . . . . .
5.18 Current vs. voltage for device D in overpumped case. . . . . . . . . . . . . . . . . . . 5.19 Gain and noise vs. voltage for device D in optimum gain case. . . . . . . . . . . . . . 5.20 Current vs. voltage for device D in optimum gain case. . . . . . . . . . . . . . . . . . 5.21 Gain and noise vs. voltage for device E in overpumped case. . . . . . . . . . . . . . . 5.22 Current vs. voltage for device E in overpumped case. . . . . . . . . . . . . . . . . . . 5.23 Gain and noise vs. voltage for device E in optimum gain case. . . . . . . . . . . . . . 5.24 Current vs. voltage for device E in optimum gain case. . . . . . . . . . . . . . . . . .
5.25 Relative conversion eciency vs. intermediate frequency for devices of di erent length. 95 5.26 Bandwidth vs. device length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.27 Output noise vs. bias voltage for device A1, with and without isolator. . . . . . . . . 5.28 Output noise vs. frequency for device A1. . . . . . . . . . . . . . . . . . . . . . . . . 96 99 100
x
5.29 Mixer noise vs. frequency for device A1. . . . . . . . . . . . . . . . . . . . . . . . . . 5.30 Output noise vs. frequency for device B. . . . . . . . . . . . . . . . . . . . . . . . . . 5.31 Mixer noise vs. frequency for device B. . . . . . . . . . . . . . . . . . . . . . . . . . . 5.32 Output noise vs. frequency for device D. . . . . . . . . . . . . . . . . . . . . . . . . . 5.33 Mixer noise vs. frequency for device D. . . . . . . . . . . . . . . . . . . . . . . . . . . 5.34 Output noise vs. frequency for device E. . . . . . . . . . . . . . . . . . . . . . . . . . 5.35 Mixer noise vs. frequency for device E. . . . . . . . . . . . . . . . . . . . . . . . . . . 5.36 Output noise vs. voltage at 6 GHz for device E. . . . . . . . . . . . . . . . . . . . . . 5.37 Output noise vs. voltage at 20 MHz for device E. . . . . . . . . . . . . . . . . . . . . 5.38 Output noise vs. frequency below 40 MHz for device E. . . . . . . . . . . . . . . . . 5.39 Output noise vs. bias voltage for device A1 in the overpumped case, for di erent LO frequencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.40 Output noise vs. LO frequency for devices A1 and B in overpumped case. . . . . . . 5.41 Measured re
ected powers for return loss. . . . . . . . . . . . . . . . . . . . . . . . . 5.42 Measured return loss for device A1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.43 Measured return loss for device A2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.44 Measured return loss for device B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.45 Measured return loss for device C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.46 Measured return loss for device D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.47 Measured return loss for device E. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.48 Measured electron temperature vs. dc power using noise thermometry. . . . . . . . . 5.49 Comparison of theoretical and experimental eciency for device A1 in optimum gain case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.50 Comparison of theoretical and experimental output noise for device A1 in optimum gain case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.51 Comparison of theoretical and experimental eciency for device A1 in overpumped case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.52 Comparison of theoretical and experimental output noise for device A1 in overpumped case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
100 101 101 102 102 103 103 105 106 106
108 108 110 112 112 113 113 114 114 119
127
127
128
128
xi
5.53 Comparison of theoretical and experimental eciency for device B in optimum gain case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.54 Comparison of theoretical and experimental output noise for device B in optimum gain case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 129
5.55 Comparison of theoretical and experimental eciency for device B in overpumped case.130 5.56 Comparison of theoretical and experimental output noise for device B in overpumped case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.57 Comparison of theoretical and experimental eciency for device C in optimum gain case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.58 Comparison of theoretical and experimental output noise for device C in optimum gain case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 131 130
5.59 Comparison of theoretical and experimental eciency for device C in overpumped case.132 5.60 Comparison of theoretical and experimental output noise for device C in overpumped case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.61 Comparison of theoretical and experimental eciency for device D in optimum gain case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.62 Comparison of theoretical and experimental output noise for device D in optimum gain case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 133 132
5.63 Comparison of theoretical and experimental eciency for device D in overpumped case.134 5.64 Comparison of theoretical and experimental output noise for device D in overpumped case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.65 Comparison of theoretical and experimental eciency for device E in optimum gain case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.66 Comparison of theoretical and experimental output noise for device E in optimum gain case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 135 134
5.67 Comparison of theoretical and experimental eciency for device E in overpumped case.136 5.68 Comparison of theoretical and experimental output noise for device E in overpumped case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Receiver noise temperature vs. rf frequency for various technologies. . . . . . . . . . 136 141
xii
A.1 A.2 A.3 A.4 A.5 A.6 A.7 A.8 A.9
for device A1 in optimum gain case. for device A1 in overpumped case. for device B in optimum gain case. for device B in overpumped case. for device C in overpumped case. for device C in optimum gain case. for device D in optimum gain case. for device D in overpumped case. . for device E in optimum gain case.
. . . . . . . . . . . . . . . . . . . . . . . . . .
148 148 149 149 150 150 151 151 152 152
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.10 for device E in overpumped case. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Tables
2.1 4.1 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Temperature pro les under weak and strong heating. . . . . . . . . . . . . . . . . . . Device geometries and dc resistances. . . . . . . . . . . . . . . . . . . . . . . . . . . . Fitted gain and noise bandwidths, optimum gain case. . . . . . . . . . . . . . . . . . Fitted gain and noise bandwidths, overpumped case. . . . . . . . . . . . . . . . . . . Device dc and ac di erential impedances in optimum gain case. . . . . . . . . . . . . Device dc and ac di erential impedances in overpumped case. . . . . . . . . . . . . . Device parameters in optimum gain case. Device parameters in overpumped case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 69 104 104 115 115 124 125
Predicted and experimental conversion eciency and output noise in optimum gain case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.8 6.1
Predicted and experimental conversion eciency and output noise in overpumped case.126 Comparison of JPL and Yale mixer results. . . . . . . . . . . . . . . . . . . . . . . . 140
xiii
List of symbols and abbreviations
HEB Hot-electron bolometer SIS Superconducting-insulating-superconducting (tunnel junction) LO Local oscillator rf radio frequency IF Intermediate frequency
Trec
Receiver noise temperature IF ampli er (input) noise temperature
TIF amp Tout Tmix Te Tb
Output noise Mixer noise temperature
Electron temperature Bath temperature Output noise due to thermal
uctuations Output noise due to Johnson noise
TT:F:
TJohn:
DSB Double side band SSB Single side band
rf
rf coupling xiv
xv
mixer IF
Mixer conversion eciency
IF mismatch factor
0 Voltage re
ection coecient
ZL Z0 RL avg
Load impedance Characteristic impedance (377
in free space) Load resistance Average dielectric constant of vacuum and substrate
D Di usion constant
Le0ph Le0e
pD 0 pD 0 Electron-electron length
Electron-phonon length
e ph
e e
l Elastic mean free path
th m
Thermal time constant Momentum relaxation time Electron-phonon interaction time Electron-electron interaction time E ective thermal time constant E ective thermal time constant t to output noise
e0ph e 0e ef f
ef f NOISE
R Resistance
RN Idc Vdc
Normal state resistance dc current dc voltage
xvi
T0 p0
normalized temperature normalized power
L Lorenz number
K Thermal conductivity G Thermal conductance c Speci c heat C Heat capacity
Gef f 0
E ective thermal conductance Electro-thermal feedback factor, no load Electro-thermal feedback factor, with load 3 dB noise bandwidth, i.e. frequency at which mixer noise doubles 3 dB gain bandwidth, i.e. frequency at which conversion eciency falls by 3 dB
fnoise3dB fgain3dB
Parameter which quanti es relative amount of di usion and phonon cooling based on device length: 0:25 (L=Le0ph )2 Electronic speci c heat per Kelvin; also magnitude of random force in Langevin equation
Chapter 1
Introduction
One of the most fundamental methods of studying physical systems is to observe electromagnetic emissions which result from changes in the system's physical state. Therefore, the development of sensitive detectors is of paramount importance, especially for astronomical and remote sensing experiments where signal levels are generally weak. This is especially true in regions of the spectrum where the earth's atmosphere is opaque, and observation time is limited to high altitude, airborne, and satellite-based telescopes. The submillimeter range of the spectrum (de ned loosely as 100 GHz to thesis. In the past ten years, remarkable progress has been made in the development of ultra-low noise receivers for the millimeter and sub-millimeter band. These receivers use as detectors superconductinginsulating-superconducting (SIS) tunnel junctions. For reasons described below, these receivers do not perform well above approximately 1 THz, and therefore there is still a need for sensitive detectors above 1 THz. As research described in this thesis will show, superconducting hot-electron bolometers are excellent candidates as ultra-low noise detectors in THz receivers. The goal of the work described in this thesis is to develop and test experimentally a model for the physical processes governing the performance of hot-electron bolometers (HEB). In this chapter, we rst motivate further the development of such a detector. Next, existing technologies already in use are described. Finally, 1
f
=
f = 3 THz) is one these regions, which is the motivation for the work described in this
2
the hot-electron bolometer is described and its advantages and disadvantages are discussed. The remainder of the thesis describes measurements to determine the sensitivity and speed of di usioncooled Nb HEB's under various operating conditions and device geometries and compares them with theoretical predictions.
1.1
1.1.1
Motivations and Applications
Emission in the Sub-mm
Emissions in the sub-mm region of the spectrum from naturally occurring objects can be divided into broadband continuum emission and emission at discrete frequencies. Broadband emission includes black-body radiation at temperatures such that
kB T
h!.
For the frequency range of interest
here, this corresponds to temperatures of order 10-100 K. While broadband radiation (such as the cosmic microwave background) is very interesting topic to study at millimeter wavelengths, there is also much to be learned by looking in more detail at the spectral intensity. Since the hot-electron bolometer is being considered primarily for use in precision THz spectroscopy applications, emissions at discrete frequencies will be considered in more detail. Sources of discrete frequencies in the submm arise from transitions between molecular vibration and rotational energy levels. Two important examples of systems which can be studied by THz spectroscopy are interstellar molecular dust clouds and the earth's atmosphere. Figure 1.1 indicates schematically the emission in the submillimeter band from a typical interstellar cloud. A fundamental outstanding problem in physics and astronomy is the birth of stars and planetary systems. Stars are born by gravitational collapse of interstellar dust in clouds, and heat must be radiated away for this collapse to continue. Much of this heat is radiated by molecular transitions in the sub-millimeter band. Information on the composition, density, and temperature of such clouds is clearly vital to the study of star and planetary formation, and can be provided by THz astronomy. THz spectroscopy has also been applied to studies of atmospheric chemistry with great success (Manney et al. 1996). Many important molecules in atmospheric chemistry such as
ClO, O3 , OH ,
and H2 0 have emission lines in the mm and sub-mm band. The rst global measurements of ClO, an
3
Figure 1.1: Schematic emission in the sub-mm band from a molecular cloud, birth place of stars and planets. Typical temperatures inside molecular clouds are 10-30 K. (From Phillips et al, 1992, with permission.)
important molecule in the destruction of
O3 , were done using mm-wave spectroscopy.
Details of the
line shape provide important information about the density, pressure, and temperature of molecules of interest. Future satellite, airborne, and balloon missions will continue these measurements on the global scale. Sub-mm emission can also be produced in arti cial quantum wells. Bloch-oscillations and Rabioscillations in semiconductor quantum-well systems have emission in the sub-mm, since the energy levels of electrons in quantum wells are typically split by a few meV.
1.1.2
Detection in the Sub-mm
At radio and microwave frequencies below about 100 GHz, low noise semiconductor based ampli ers are available to pre-amplify incoming signals before detection and further signal-processing (Maas 1993). At higher frequencies, incoming signals are coupled directly to a detector. Therefore, the noise performance of the detector dominates the system noise and is thus deserving of the most
4
attention. Generally, there are two modes of detection to consider:
coherent incoherent
or
direct
detection, and
or
heterodyne
detection. In direct detection mode, a dc signal is generated across the
detector which is proportional to the rf power coupled to the device over a very wide frequency range. (The phase of the incoming signal is not preserved, hence the term incoherent.) Therefore, a device operating in direct detection mode o ers essentially no spectral resolution. To get spectral resolution, the rf radiation must be ltered before it is detected. Fourier transform spectrometers in the sub-mm band can be built for this purpose, but the spectral resolution is at best 1=
1005 (Harris
1990). For identi cation of single molecular species, a spectral resolution of at least 1=
1006
is required. Therefore, heterodyne detection is required for spectral line studies. Additionally, the hot-electron bolometers developed in this thesis are intended for use in heterodyne systems, so more attention will be devoted to such systems. Heterodyne detection requires a non-linearity in the detector which \mixes" the signal with a local oscillator. For this reason, the detector is also referred to as a
mixer.
Figure 1.2 indicates a
schematic of heterodyne detection. The (weak) signal to be detected and a locally produced coherent signal (the local oscillator, LO) are applied to the detector. Non-linearity in the detector produces an output signal at the di erence or
intermediate
frequency (IF), which is then ampli ed, typically
by a cooled, low-noise GaAs based HEMT transistor and then detected by a spectrum analyzer. The phase of the IF signal is equal to the phase of the incoming signal (plus a constant), hence the term coherent detection. The parameter used to quantify the noise of a given heterodyne system is the power per frequency at the rf input required to give a signal to noise of one at the output intermediate frequency. Therefore, the units are W/Hz. Since a (matched) resistor at physical temperature T radiates kB T power per frequency in the Rayleigh-Jeans limit, a common unit for noise power is noise This is the temperature a matched resistor would be at to radiate
temperature.
kB T
power per frequency into
the receiver. (This does not necessarily correspond to the temperature of any physical system.) The receiver noise temperature is related to the various components of gure 1.2 as follows:
Trec:
=
Tout rf mixer
+
TIF amp: : rf mixer IF
(1.1)
5
Antenna fRF
Mixer
Low noise amplifier fIF= fLO-fRF
Spectrum analyzer
fLO Local Oscillator
Figure 1.2: Schematic of a heterodyne detection system. For frequencies below 100 GHz, a low-noise pre-ampli er would be used between the antenna and the mixer.
Here
Tout
is the output noise generated by the mixer itself,
TIF amp:
is the
input
noise of the IF
ampli er, rf is the input (rf) power coupling to the mixer, IF is the coupling between the mixer and the IF ampli er, and
mixer
is the conversion eciency of the mixer, de ned as the ratio of
the available power at the mixer IF output to the available signal power at the mixer rf input. IF ampli ers with noise temperatures in the range of 2
0 3 K are readily available. It should be clear
the the mixer output noise and conversion eciency are crucial parameters in the system. The rf and IF coupling can be made unity at select frequencies by careful circuit design, if the input and output impedance of the mixer are known. In order to quantify the contribution of the mixer itself to the system's overall noise, independent of the coupling circuit, the
mixer noise
is de ned as
Tmix
Tout mixer
(1.2)
For a given device, this is the minimum receiver noise temperature, if the coupling circuits are perfect and the IF ampli er generates little noise in comparison with
Tout .
Clearly the mixer noise is the
most important device property, but there are other practical considerations that de ne desired mixer properties. In addition to having a low noise, it should
1) require low LO power, 2) be fast
6
3) be easy to couple to at both rf and IF frequencies.
Local oscillators at THz frequencies are usually xed in frequency, and have limited power, typically not more than a few mW at best. Therefore, a mixer which requires low local oscillator power is desirable. During observation, the local oscillator frequency is xed, and the IF output of the device is sent into a spectrum analyzer, with many GHz of bandwidth. The spectrum analyzer (usually an acousto-optic spectrometer) is designed to measure all frequencies within its band simultaneously. The signal-to-noise in each frequency bin depends on the integration time at each bin, and therefore the most ecient use of the available observation time is to measure as many di erent frequencies as possible simultaneously. For this to be possible, the detector should be fast enough to allow detection of signals within several (10-20) GHz of the local oscillator. If the local oscillator could be swept, and the intermediate frequency were kept at some low value, say 1 MHz, then the system would be \viewing" only a 1 MHz bin of the signal at a time. A broader band IF system allows the device to view a larger slice of the signal spectrum for a given LO frequency, and results in a larger observation/integration time per rf frequency bin. At THz frequencies, the rf coupling is usually achieved quasi-optically. A Si or quartz lens focuses an incoming THz (Gaussian) beam onto a micro-fabricated antenna, which can be a dipole, slot, bowtie, spiral, or log-periodic antenna, for example. The antenna is a coupling circuit and transformer in one. Since the coupled device is much less than a wavelength in size, it can be treated as a lumped circuit element. The e ect of the antenna is described as a voltage source with a particular source impedance. The source impedance is usually of order
Z0 =pavg (=
105
for Si), where avg
is the average dielectric constant of vacuum and the dielectric lens (Rebeiz 1992; Rutledge et al. 1983; Elliott 1981), and
Z0
= 377
is the characteristic impedance of free space. For example,
the source impedance of self-complimentary antennas (which have broadband frequency response) is 114
on quartz (Rebeiz 1992). In order to achieve maximum coupling between the device and the incoming beam, the device impedance should be equal to the complex conjugate of the source impedance. 100
. The device output impedance should be well-matched to the IF ampli er. The input impedance of most \o the shelf" IF ampli ers is 50
, by convention. From this point of view, an output
Broadband matching to the device is easiest if the device impedance is real and of order
7
impedance of 50
is most desirable for a mixer. In principle, IF ampli ers can be designed with an input impedance that is di erent from 50
. The ideal system would have a mixer output impedance equal to the input impedance of the HEMT (the rst stage of the IF ampli er), since this would require no matching network between the device output and the HEMT gate input. However, since low-noise ampli ers with 50
input impedance are readily available, a mixer output impedance near 50
is the best compromise. So far, we have considered an input at only one signal frequency (i.e. single-side-band operation). However, it should be clear that two di erent signal frequencies, if symmetrically placed above and below the local oscillator frequency, will cause an IF output at one frequency, given by the absolute value of the di erence between the LO and rf frequency. More sophisticated circuits can be constructed which are sensitive to only one \side" of the local oscillator, the \upper side band", or \lower side band." Engineers must decide whether to build the more complicated circuit, or to keep the circuit simple and use other methods to determine whether a particular feature in the intermediate frequency spectrum arose from the upper-side-band, or lower-side-band, such as adjusting the LO frequency and tracking which way the peaks move. If the system is being used to detect signals in both the upper-side-band and the lower-side-band at the same time, then the important noise parameter is the
double side band
noise temperature.
plus
This is de ned as the signal power per frequency at the rf input in the upper-side-band
the
signal power per frequency at the rf input in the lower-side-band required to give a signal to noise of one at the output intermediate frequency. The power per frequency at the rf input in
one single-side-band
noise temperature is simply that
of the side-bands required to give a signal-to-noise of
one at the output intermediate frequency. In most systems, the sensitivity to the upper-side-band is equal to the sensitivity of the lower-side-band, and therefore the noise temperatures are related by:
Trec (DSB ) = Trec(SSB )=2:
(1.3)
Classically, there is no lower-limit to the noise temperature of a receiver. However, quantum mechanically this is not the case. The mixers considered in this thesis can be classi ed as phaseinsensitive linear ampli ers, since the system gain is independent of the phase of the rf input signal, and the output power at the IF is proportional to (and larger than) the input power at the rf.
8
Regardless of the details of the system, it can be shown on very general grounds (Caves 1981) that the minimum noise temperature of such an ampli er is given by Tmin = h!=kB ;
(1.4)
where
is a number of order unity. The noise temperature must be very clearly de ned when
operating near this noise
oor, and for the various possible de nitions,
= 1 ,
2,
ln2,
or
ln3 (Tucker and Feldman 1985). Since the experiments described in this thesis do not approach this noise
oor, we will not discuss the issue further and take
= 1. This noise can be considered
a combination of quantum
uctuations in the input signal as well as quantum
uctuations in the internal degrees of freedom of the measurement system (i.e., the mixer). As long as the ampli er is linear and phase-insensitive, the details of the internal degrees of freedom always conspire to enforce equation 1.4. This limit is referred to as the \quantum limit". The goal of any research on THz mixers is to construct a device which achieves this limit of sensitivity and ful lls the three practical requirements mentioned above. As we will see, HEB's promise to achieve these goals.
1.2
Existing technologies
There are two types of mixers in broad use at THz frequencies: Schottky diodes and SIS tunnel junctions. Both are brie
y discussed here.
1.2.1
SIS tunnel junctions
SIS tunnel junctions are widely used as ultra-sensitive mixers for frequencies between 100 GHz and 1 THz. The material of choice is Nb, which has a critical temperature of 9.25 K. The energy gap in this material, 3 meV, corresponds to 740 GHz. Experiments performed during the 1980's demonstrated that the quantum limit of sensitivity, eq. 1.4, could be achieved using SIS detectors (McGrath et al. 1981; Face et al. 1986; Mears et al. 1990). Today, practical systems utilizing SIS mixers are in wide use, and typically achieve a noise temperature a few times the quantum limit. SIS detectors have a geometric capacitance which necessitates careful design of rf coupling circuits. Thus, SIS tunnel junctions are
low noise, fast, and require low LO power.
However, they are
not
9
easy to match at IF and RF frequencies.
Additionally, SIS detectors have degraded performance
above the gap frequency, and are expected to degrade sharply at twice this frequency. Highergap superconductors such as NbN or even high-Tc superconductors may achieve quantum-limited performance at frequencies above 1 THz, but none have been demonstrated so far. Finally, the geometric capacitance requires the construction of low-loss tuning circuits which are dicult to construct at frequencies above the gap frequency of Nb.
1.2.2
Schottky diodes
Schottky diodes are used as mixers for frequencies above 1 THz. The noise performance of Schottky diodes is typically no better than 150 times the quantum limit (Crowe, Mattauch, Roser, Bishop, Peatman, and Liu 1992). Additionally, Schottky diodes require large LO power, mW, necessitating the use of bulky molecular gas lasers as local oscillators. Schottky diodes are
low noise, and require large LO power. fast,
but they are
not
1.3
Hot-electron bolometers
hot-electron
A bolometer is a detector that functions on the following simple principle: rf signal power heats a device, and this temperature shift is measured via an appropriate thermometer. In a
bolometer, the electrons are heated by dc and rf power above the temperature of the lattice. If the resistance of the electronic system depends on the electron temperature, then the resistance can be used as a thermometer. A bolometer can operate as a mixer since the temperature shift is proportional to the power absorbed:
T
/
V (t)2 =R
VLO cos(!LO t) + Vsig cos(!sig t)
LO sig
2
VLO Vsig
0 1 cos (! 0 ! )t + dc and high frequency terms: 0!
(1.5)
Since the temperature changes at the intermediate frequency (!LO
sig ), the resistance changes
at the intermediate frequency. Under a current bias, this leads to an oscillating voltage at the intermediate frequency. Thus, the device performs as a mixer. This process will be analyzed in detail in later chapters. Here it is important to note the intermediate frequency must be less than
10
the energy-relaxation rate for the electron system, otherwise the electron temperature will be unable to follow the di erence frequency. Quantitatively, the mixer conversion eciency drops by 3 dB at an intermediate frequency given by
f3dB = 1=(2 th);
(1.6)
where th is the thermal (cooling) time constant. For the purposes of this thesis, this frequency will be de ned as the
gain bandwidth.
It is this issue that has limited the use of hot-electron-bolometer
mixers, and which research in this thesis addresses in a very direct way.
1.3.1
Semiconductor based HEBs: InSb, AlGaAs
The rst hot-electron-bolometer mixers used electrons in lightly doped n-type bulk InSb. When the InSb is cooled to 4 K or below, the electrons become decoupled from the lattice and hence can be heated above the lattice temperature. The mobility and hence the resistance depends on the temperature (
/T
e
3=2
), so the semiconductor can be used as a mixer. The results of the rst mixing
experiments in InSb were published in Arams et al. (1966). The rst practical receiver was described in Phillips and Je erts (1973). Since then, receivers have achieved noise temperatures of 20 to 25 times the quantum limit at rf frequencies of 500, 600 and 800 GHz (Brown et al. 1985; Padman et al. 1992). Higher rf frequency operation is more dicult since the bulk InSb sample would have to be integrated into an antenna. (The above results utilized waveguide coupling.) Additionally, higher frequency operation based on resonant absorption at the cyclotron frequency (as in Brown et al. (1985)) requires a magnetic eld up to 1 Tesla. The main reason that InSb receivers are not in widespread use is the limitation on the maximum intermediate frequency, given by the electron-phonon inelastic interaction rate. In InSb, this rate is
(0:1s)01 , which limits the intermediate frequency bandwidth to about 1 MHz. This is usually
low
unacceptably low. A multi-GHz intermediate frequency bandwidth is desired. InSb mixers are
noise,
but
slow.
The two dimensional electron gas (2-DEG) in modulation-doped AlGaAs/GaAs heterostructures has also been investigated as a potential medium for use as hot-electron mixers (Yang et al. 1995; Yang et al. 1993). By applying enough LO and dc power to heat the electrons in a quantum well to approximately 90 K from a lattice temperature of 4.2 K or 19 K, it was possible to observe
11
mixing and measure the thermal time-constant. A gain bandwidth of 1.7 GHz was measured. Lownoise receivers based on the 2-DEG medium have not yet been demonstrated, and would have to overcome some obstacles. The operating temperature of the electrons must be above approximately 80 K, because the mobility is only weakly temperature dependent below this temperature. This will increase the noise compared to lower temperature devices. Additionally, due to the high mobility the momentum-relaxation time m is long enough to cause
! m > 1 at THz frequencies.
This will
cause the rf (THz) impedance of the 2-DEG to develop an inductive component, decreasing the absorption of THz power and hence decreasing the device eciency, or bandwidth if a tuning circuit is designed to compensate the kinetic inductance.
1.3.2
Superconductor based HEBs: Nb, NbN
Tc
can be used as a very sensitive bolometer, since the resistance depends
A superconductor near
sharply on temperature. In dirty metal thin lms below about 10 K, the electron-electron interaction rate is much faster than the electron-phonon interaction rate, so that dc and rf power can heat the electrons above the temperature of the lattice. Additionally, in dirty metals, the electron momentum relaxation time is very short (< 1 fs), so that power should be absorbed up to very high frequencies,
100 THz. Gershenzon et al. (1982) showed that dirty Nb superconducting thin lms absorb
radiation uniformly from 1010 to 1015 Hz, with approximately the same eciency. There, a magnetic eld as well as a current bias was applied to bring the lms into the \resistive" state near frequency radiation was applied, and the change in the resistance was measured. Nb superconducting HEB's were rst used in mixing experiments in 1983. In these experiments (Gershenzon et al. 1983; Gershenzon et al. 1988; Gershenzon et al. 1989; Gershenzon et al. 1990; Gershenzon et al. 1990), the electron-energy relaxation rate in Nb (the electron-phonon interaction rate) was measured for a variety of samples at temperatures from 1.6-10 K. It was found that the electron-phonon interaction time was of
Tc.
High
1 ns at 4.2 K for dirty lms, with a di usion constant
D
= 1
cm2 =s.
This would allow for an intermediate frequency bandwidth of
150 MHz,
which is still too small for practical applications. Additional theoretical modeling (Gershenzon et al. 1990) suggested that the mixer noise temperature could approach 50 K, which is the quantum limit at 1 THz. This prediction was independent of the rf frequency, as long as the rf radiation
12
was absorbed by the electron system. This is true up to the electron-momentum relaxation rate,
100 THz (Prober 1993). Thus, the noise was predicted to be low up to very high rf frequencies,
but the IF bandwidth was not sucient. Two approaches have been proposed to increase the intermediate frequency bandwidth of the superconducting bolometer, while keeping the noise low and the rf frequency range broad. The rst approach is to use a material with a shorter electron-phonon interaction time. NbN has a somewhat higher
Tc
than Nb, and therefore also a stronger electron-phonon interaction. The predicted
noise is still low, and the rf frequency range should also be broad. Initial experiments indicated an IF bandwidth of 5:3 GHz (Gol'tsman et al. 1991; Gousev et al. 1994). There, results between 1.6 K and 5.3 K were presented. By extrapolating the data to 10 K, the authors claim a bandwidth of 10 GHz could be achieved. Subsequent experiments have been unable to reproduce these results. The results have varied for the intermediate frequency bandwidth (0:6 GHz (Dzardanov et al. 1994), 1:1 GHz (Gol'tsman et al. 1995), 0:8 GHz (Karasik et al. 1995; Ekstrm et al. 1995), o 3
0 4 GHz (Yagoubov et al.
1996), 1:6 GHz (Kawamura et al. 1996), 2:2 GHz (Kawamura et al.
1997).) For some lms apparently comparable to those of Gousev et al. (1994), the mixing bandwidth was less than 1 GHz. Recent experiments (Yagoubov et al. 1996) indicate that control of the lm thickness may allow more control over the achieved bandwidth for NbN. Very thin lms (3.5 nm) achieve the largest bandwidths. Promising sensitivities have also been achieved, between 410 K (DSB) at an rf frequency of 410 GHz, and 9000 K (DSB) at 1.2 THz. Thus NbN is worthy of further investigation. A di erent approach was proposed by Prober (1993), and is the approach investigated in this thesis. The approach consists of using a very short strip of Nb as a hot-electron bolometer, less than the electron-phonon interaction length, interaction rate and
Le0ph
pD 0
e ph , where e ph is the electron-phonon
01 0
D the di usion constant.
The electron-phonon interaction length is essentially
the length which an electron di uses before emitting a phonon. For short bridges, cooling of the electrons can occur by out-di usion of heat into high-conductivity, normal metal leads. In this case, the thermal time constant is the di usion time, given by
dif f
=
L2 : 2 D
(1.7)
13
(The numerical factor will be derived in chapter 2.) Thus, for a 0:1 constant of 1
m
bridge with a di usion
cm2 =s, a time constant of order 10 ps is predicted, allowing an intermediate frequency
bandwidth of order 10 GHz to be achieved.
1.4
Thesis overview
The research in this thesis was designed to determine the limits on the speed and sensitivity of \di usion-cooled" hot-electron bolometers. In chapter 2, an overview of the expected theoretical performance is presented, and the expected validity of the theories is discussed, with suggestions for future modi cations of the theory. In chapter 3, the experimental designs and techniques are further discussed. The device geometries, dc properties, and fabrication techniques are discussed in chapter 4. Due to the relative ease of experiments below 100 GHz, mixing experiments were performed on devices of di erent lengths with rf and LO frequencies of 10-40 GHz. The results will be presented in chapter 5. The thermal time-constant is found to obey the scaling law predicted by equation 1.7. Additionally, the noise and conversion eciency were measured, as a function of LO and dc power and intermediate frequency, for devices of di erent lengths at a bath temperature of 2 K. The output noise is found to be dominated by thermal
uctuation noise, whose frequency dependence is similar to the frequency dependence of the conversion gain. This result increases the available intermediate frequency bandwidth by a factor of 1.4-9.4. In chapter 6, the results are put into a broader context by comparing with other results on di usion-cooled bolometers. Then, in chapter 7, conclusions are drawn. The hot-electron bolometer can be fast, with a bandwidth of at least 6 GHz, and is very sensitive. The measured mixer noise is between 120 K and 530 K (DSB). Thus, di usion-cooled hot-electron mixers can simultaneously achieve very low noise and very large bandwidth, and promise to dramatically improve the noise performance of THz receivers.
Chapter 2
Theory
We begin this chapter with a generic description of bolometers as mixers, and then the particular realization for Nb hot-electron bolometers is discussed, rst as a lumped element and then using a distributed element approach. The important quantities of interest are the conversion eciency and noise as a function of intermediate frequency, dc power, LO power, and temperature. The device geometry will also be discussed, as it is important for di usion-cooled bolometers.
2.1
Mixing in bolometers: generic description
In this section, the conversion eciency and noise for hot-electron bolometers is derived using a pedagogical approach. More rigorous derivations can be found in the cited literature. Although we are only considering the case of hot-electron bolometers, the expressions are valid for any generic bolometric mixer. There are two intrinsic noise sources that are important for hot-electron bolometers: Johnson noise and thermal
uctuation noise.
Quantum
noise will be an important noise source
in the future if the other noise sources are made small. This will be discussed at the appropriate times in the following sub-sections. In the mixers described in this thesis, thermal
uctuation noise dominates. In the rst sub-section, the conversion eciency will be calculated. Next, the thermal
uctuation noise will be calculated, in the absence of electro-thermal feedback. Then, the total noise at the output in the presence of electro-thermal feedback is calculated. The nal subsection 14
15
discusses the
mixer noise.
2.1.1
Calculation of conversion eciency
In bolometric mixers, the non-linearity which causes mixing between the LO and signal (rf) frequencies is the fact that the resistance change is proportional to the power, which is proportional to the voltage
squared.
This concept is indicated schematically in gure 2.1. The LO voltage and signal
voltage are shown before they are combined. The signal voltage is usually much smaller than the LO voltage. The voltages are added and then applied to the device. (This can be achieved optically, for example, in a 50/50 beam splitter before the gaussian beam is focused onto the device.) The instantaneous power dissipated is given by
V (t)2 =R.
The electron temperature cannot follow the rf
and LO frequencies. However, if the di erence frequency is small enough, then the electron temperature is modulated at the di erence frequency. This causes a resistance change at the di erence frequency, and hence a voltage drop across the device at the di erence frequency. In order to describe this process quantitatively, we write down a di erential equation to describe the electron temperature as a function of time in the case of a pure current bias at the dc and IF frequencies. This will allow the development of an intuitive way to describe the mixing, as well as introduce the concept of electro-thermal feedback. In a later section, the assumption of current bias at the IF will be relaxed. We assume the dissipated power oscillates at the di erence frequency, and write:
2 Pinput (t) = Idc R(Te (t)) + Pac ei!IF t ;
where
!IF
! 0!
LO
rf ,
Pac
= 2
pP
(2.1)
LO rf (from equation 1.5), and
P
PLO , Prf
are the LO and
rf powers, respectively. (There will also be rf oscillations in the input power which the electron temperature will not be able to follow.) The LO and rf power will cause a dc component of the input power, which is not included because the response at the IF is of interest here.
R(Te (t)) is the
resistance, which depends on temperature, which in turn depends on time.1 The power
owing out is (in the linear response regime) proportional to the temperature di erence between the bolometer
1 At this point, no non-linearities in the underlying I-V curve are being postulated. The mechanism for the R(T) curve is not described at all; it is simply a phenomenological input parameter of the theory.
16
1
Voltage (arb. units)
LO voltage 0 RF voltage -1 Time (arb. units)
Sum voltage (arb. units)
1
0
-1 Time (arb. units)
Power (arb. units)
1.0
0.5
0.0 Time (arb. units)
Temperature (arb. units)
2
1
0 Time (arb. units)
Figure 2.1: Schematic of the mixing process in bolometers. The electron temperature cannot follow the high-frequency signal, but it can follow the intermediate frequency. The parameters used to generate this plot are: !IF = 0:1 !LO , VRF = 0:1 VLO .
17
and a thermal reservoir or \bath":
Pout (t) = G(Te (t) 0 Tb ):
Here
(2.2)
Te is the bolometer temperature, which will be the electron temperature when we consider the Tb is the bath temperature.
The electron temperature
special case of hot-electron bolometers below.
is time dependent, but the bath temperature is held xed. (In the more general case, the expressions derived below based on equation 2.2 are still valid if G is replaced by
dPout =dTe (Keizer 1987).)
The
in
owing power can either heat up the electrons or
ow out to the bath. The conservation of energy then takes the form:
Pinput (t) = dE (t)=dt + Pout (t) = CdTe (t)=dt + Pout (t):
Here,
(2.3)
C
=
dE=dTe ,
with
E
the (thermal) energy stored in the electron system. (C is assumed to
be independent of temperature in this calculation.) The electron temperature will have a dc shift above the bath temperature, as well as an oscillating component. Therefore, we make the ansatz:
Te (t) Te (t) 0 Tb = Tdc + Tacei!IF t :
We can now solve for the temperature by Taylor expanding
(2.4)
R(T ) as
(2.5)
R(Te (t)) R(Tb ) + dR=dTe 3 (Te (t) 0 Tb ) = R(Tb) + dR=dTe 3 Te (t);
and substituting into equation 2.3 expression 2.4 for the temperature of the system. This results in2
Tdc Tac
= =
Pdc Gef f Pac Gef f
(2.6) 1 + i!IF ef f 1 (2.7)
2 To include the e ect of the rf and LO power on the dc shift in the electron temperature, P should be replaced dc by Pdc + PLO + Prf .
18
where
ef f th Gef f 0
th =(1 0 0 ); C=G; G (1 0 0 ); 2 Idc dR=dT : G Pac =G
(2.8) (2.9) (2.10) (2.11)
The electron temperature will be modulated by
if the intermediate frequency is small. As
the intermediate frequency increases, the ac component of the electron temperature decreases. The e ect of the electro-thermal feedback between the electron temperature and the dc bias supply is described quantitatively by the parameter
0 .
If
0
is small (due to small current or In particular, the e ective and the e ective thermal
small dR/dT), then the e ect of electro-thermal feedback is small. time constant conductance
ef f
is equal to the \bare" thermal time constant is equal to the bare thermal conductance
th ,
Gef f
G.
If the parameter
0
is large, then
\electro-thermal feedback" is strong. There is a simple explanation for this: under a current bias, a small increase in the resistance causes the dc power dissipated to increase, since
Pdc
=
I 2 R.
An increase in the input power causes the system temperature to rise. If dR/dT is positive, as is the case for superconducting bolometers, increasing the temperature serves to further increase the resistance, and there is positive feedback. Thus,
0 is a quantitative measure of this feedback:
when
0 = 1, there is thermal runaway.
is reached, and then dc power supply is a
In practice, the temperature will increase until the normal state will be zero, stopping the process. On the other hand, if the
dR=dT
and
0
voltage bias,
then it will serve to stabilize the system temperature: an increase in the power, since
in the resistance will cause a
decrease
Pdc = V 2 =R, which will cause a decrease
in the system temperature, which will cause a decrease in the resistance, opposing the postulated resistance increase. Note that this will have the same e ect on
uctuations: thus, electro-thermal feedback a ects the system's response to an external stimulus as well as internal
uctuations of the temperature, enhancing both in the case of a current bias, and suppressing both in the case of a voltage bias3 .
3 If dR/dT is negative (as in InSb), then the conclusions are reversed: A current bias will tend to stabilize, while a voltage bias will destabilize.
19
The ac voltage at the intermediate frequency is given by:
Vac = Idc dR=dT Tac = Idc dR=dTe
Pac 1 : Gef f 1 + i!IF ef f
(2.12)
The conversion eciency is de ned as the ratio of the available output power at the IF to the available input power at the RF. The output power at the IF is proportional to
2 Vac .
The above
calculation was done assuming no load at the IF. If the IF load is equal to the bolometer resistance, then the AC voltage calculated above will be divided in half. The power coupled into the ampli er will be (1=2) Vac 2 =R, where Vac is given by one half the value of equation 2.12. Thus, the conversion eciency is given by
j j
(!IF ) =
PLO Idc (dR=dT ) 2R Gef f
!
2
1 + (!IF ef f )2
1
:
(2.13)
From this equation, it is clear that the conversion eciency drops by 3 dB at an intermediate frequency given by equation 1.6, with th replaced by the more general ef f :
f3dB = 1=(2 ef f )
In a later section, the physical processes which govern th will be discussed in more detail.
(2.14)
The above calculation assumes that the current bias applies at both dc and at the intermediate frequency. However, at the intermediate frequency there is an additional load resistance, given by the input impedance of the intermediate frequency ampli er. If the intermediate frequency is in the GHz range, the ampli er input impedance is usually on the order of 50
. A more careful analysis of the electro-thermal feedback taking into account the di erent environmental impedance at the dc and IF frequencies shows that the e ect of the load resistance is to modify et al. 1966; Mather 1984; Karasik and Elantiev 1996):
0 as follows (Arams
2 Idc dR=dT RL 0 R G RL + R
=
0
R 0 R ;
RL + R
L
(2.15)
where RL is the load resistance at the IF, i.e. the input resistance of the IF ampli er. This de nition of
reduces to the previous de nition when RL ! 1.
Since the load resistance is typically 50
,
if the device impedance is comparable to this, the electro-thermal feedback will be signi cantly
20
suppressed, according to 2.15. Physically, this is because the device is no longer \seeing" a current source at the intermediate frequency, but a 50
\shunt". This is more like a voltage source than a current source, which tends to suppress electro-thermal feedback, as discussed above. In addition to modifying the electro-thermal feedback, there will also be an impedance mismatch factor between the device output impedance and the input impedance of the IF ampli er, which should multiply equation 2.13 to predict the and Sard 1966). It is given by:
coupled
conversion eciency (Arams, Allen, Peyton,
IF
=
(R + RL )2
4RRL
:
(2.16)
This factor is not actually a standard mismatch factor in the usual sense, since the device impedance is not independent of frequency whereas equation 2.16 is. The factor results from a more rigorous calculation of the e ect of a nite load impedance at the intermediate frequency on the electron dynamics (Karasik and Elantev 1995). The parameter varies between zero and one, and is one when the device output impedance is equal to the input impedance of the IF ampli er. Finally, it should be noted that the resistance in the above calculations was assumed to depend only on the electron temperature, and not on the current. This is not necessarily true
a priori
for
any bolometer, including those studied in this thesis. However, the deviations from linearity are generally dicult to predict theoretically, so the standard approach is to neglect the dependence of the resistance on the current, assume it is not important, and see how well this assumption is validated by experiments. This will be discussed further in later sections.
2.1.2
Thermodynamic temperature
uctuations: Langevin approach
In this subsection, we will calculate the
uctuations in the electron temperature using a Langevin equation approach, where a stochastic forcing function is applied to the system4 . Any thermal system in equilibrium with a bath exchanges energy with that bath on a very rapid time-scale. Thus, the mean energy of the system will be constant, but it will
uctuate in time about the mean value. The energy
uctuations will give rise to temperature
uctuations, which cause resistance
uctuations if dR/dT is non-zero and hence voltage
uctuations if there is a current bias.
4 This discussion is not rigorous but illustrates the important physical concepts. A rigorous presentation would require discussion of the Wiener-Khintchine theorem.
21
Consider the
ow of energy, when no external power is applied. Then:
Pinput (t)=C
= =
G (T 0 T ) C e b dTe =dt + ( th )01 Te = f (t): dTe =dt +
(2.17) (2.18)
If no external power is applied, there is still a random exchange of energy with the bath. This is modeled by postulating a forcing function f(t) which is random, whose auto-correlation function is given by:
< f (0)f (t) >=
(t):
Here the brackets
(2.19)
< : : : > denote an ensemble average, which is equivalent to a time average if the
system is stationary. The strength of the forcing function is given by
. According to equation 2.18, the time evolution of the electron temperature is no longer a deterministic process. However, there is a mean and variance of the electron temperature, and these can be calculated from the theory of Langevin and Fokker-Planck equations (Gardiner 1983). The result, for equation 2.18, is
< Te > < (Te )2 >
=0 = 1
th 2
(2.20) (2.21)
Result 2.21 is sometimes referred to as the
uctuation-dissipation theorem (Keizer 1987). It relates the magnitude of the forcing function (
), the variance of the system (< (Te )2
>),
and the
\dissipative" term, ( th )01 . Usually, the dissipative term describes real dissipation. For example, in Brownian motion, the equation of motion for a particle's position is given by an analog of equation 2.18, namely
dv=dt + v = f (t):
Here, it is clear that
(2.22)
describes real friction.
(The
uctuation-dissipation theorem for equation 2.22 This does not allow the determination
would relate a random force to the friction
and < (v)2 >).
of the variance of the electron temperature. For this, we need to insert a result from thermodynamics. We can calculate
< (Te )2 > using the Boltzmann distribution, namely that the probability for the
22
system to be in a state with energy E is proportional to
e0E=kB T .
Thus (Kittel 1980):
< (Te )2 >=
kB Te2 C
(2.23)
From this expression, and 2.21, the magnitude of the forcing function can be calculated, and then the spectrum of the thermal
uctuations can be calculated. One nds
=
kB Te2 th C
2
(2.24)
Now consider the spectrum of the
uctuations of the electron temperature (Van der Ziel 1976). We make the ansatz:
Te (t) =
X T (!)e
e !
i!t
; f (t) =
X f (! )e
!
i!t
:
(2.25)
Now Te (! ) and f (! ) are random variables. Upon substituting this into equation 2.18, one nds a re-
3 lationship between f (! ) and Te (! ). Since the spectral density of Te (! ) is given by 2Te (! )Te (! ),
and since the spectral density of
f (!) is white, one nds
Te (!)
2
=
2 4 kB Te
G
1 + (! th )2
1
:
(2.26)
This is the central result of this sub-section. If the temperature
uctuates, then so does the resistance. If there is a current bias, there will be a voltage
uctuation, hence the term \thermal
uctuation noise." The
uctuating voltage can be calculated, and is found to be
V (!)
2
=
Idc dR=dTe
4k
2
B e
T2 1 : G 1 + (! th )2
(2.27)
The assumption of a white noise source is equivalent to the assumption of random energy exchanges with a reservoir. Since, for most hot-electron bolometers, the energy exchange is via
phonons, thermal
uctuation noise is sometimes referred as \phonon noise". It is assumed that the phonons are uncorrelated and are emitted and absorbed on a very fast time-scale by the electron
0 system, much faster than th1 . If this is not the case, then the above theory would need to be
modi ed.
23
If the output impedance of the device is equal to the input impedance of the IF ampli er, then the ac noise voltage is also halved. In that case, by calculating the power coupled into the device, we can determine the output noise temperature of the device due to temperature
uctuations. This method yields:
Pout kB Tout 1B = Idc Te (dR=dTe )
where 1B is the bandwidth.
2
1 kB RG 1 + (! th )2
1B;
(2.28)
In the case of Brownian motion, there is a quantum analog of equation 2.22 (Callen and Welton 1951; Gardiner 1991). In that case, the position and momentum of the particle are interpreted as quantum mechanical operators. Thus, there is also a quantum
uctuation-dissipation theorem. The quantum
uctuation-dissipation theorem is usually referred to as \the"
uctuation-dissipation theorem. The quantum result is equivalent to the classical result so long as kB T
> . h!
If this is not
the case, quantum noise must be considered in the motion of the particle. This can be done, and in that case the forcing function on the R.H.S. of equation 2.22 is replaced by a more complicated operator expression (Gardiner 1991). The fact that a quantum treatment can give results that are equivalent in certain limits to the classical case can be viewed as profound and interesting, or as just another example of the Bohr correspondence principle. Later it will be important to discuss the ultimate limits of the bolometer. Classical predictions, based on the
classical
Langevin equation 2.18 can predict a mixer noise temperature less than the
\quantum limit", equation 1.4. This is because the classical calculation outlined above does not take into account quantum
uctuations in the resistance or temperature of the electrons. In order to quantify these quantum
uctuations, a quantum Langevin approach will be necessary. Unfortunately, the temperature is not an operator in quantum mechanics. However, it should be possible to de ne an
energy
operator, and calculate the energy
uctuations in a fully quantum mechanical way. The
formalism for this approach has been developed (Gardiner 1991), but is not usually applied to the energy of a system. This is because in most experiments demonstrating quantum e ects, the energy is not an observable. However, to the extent that the electron resistance is dependent on the energy of the electrons, the energy theoretical work is merited.
is
an observable in hot-electron bolometer experiments, and so further
24
The above approach has taken a top down view of the problem. However, a more microscopic theory can be investigated, and has been to a certain extent (Kogan 1996). In a more microscopic theory, a Boltzman equation approach is taken, where the occupation probability is calculated for a position in phase space at a given time. A
uctuation term can be added to the scattering
term in the Boltzman equation, as in equation 2.18 for the electron temperature. Again, that treatment is semi-classical, and does not allow for quantum
uctuations to be accounted for within the theory. Recently, Kogan (1991) developed a
quantum Langevin-Boltzman
equation, generalizing
the Keldysh non-equilibrium approach to include, in a quantum mechanical manner,
uctuations in the occupation probability. However, this approach has not been applied to the problem of thermal
uctuation noise in electron bolometers. To do so would require calculating how quantum
uctuations in the occupation probabilities of regions in phase space translate into
uctuations in macroscopic observables, such as energy, temperature, or resistance. There is hope for a quantum treatment of thermal
uctuation noise, but none has yet been given. Thus, in this thesis, the thermal
uctuation noise will be treated classically. In the midst of theoretical confusion, this much is clear:
A full quantum treatment of hot-electron bolometers will not predict a mixer noise below the quantum limit of equation 1.4.
Therefore, as far as the development of detectors goes, the easiest way to build
a quantum-noise limited detector will be to nd a detector whose classical prediction for the noise is below the quantum limit. This is because quantum mechanics usually predicts more noise than a classical calculation. Let the reader beware, however, that, this is not a theorem. There are (at least) two counter-examples: sub-shot noise in mesoscopic microbridges (de Jong 1995; Steinbach et al. 1996) and sub-shot noise in quantum optics (Mandel and Wolf 1995).
2.1.3
E ects of electro-thermal feedback on output noise
Above, the
uctuating ac voltage generated by a current bias was calculated with no load at the intermediate frequency, and in the absence of electro-thermal feedback. In the presence of electrothermal feedback, and an ampli er load
RL ,
an educated guess as to the form of the thermal
uctuation noise would be a modi cation analogous to the modi cation of the conversion eciency. This is because, as discussed above, electro-thermal feedback serves to enhance or suppress both
uctuation and the response to an external stimulus in the same way. The prediction for the
25
(coupled) output power in the presence of electro-thermal feedback is (Mather 1982; Karasik and Elantev 1995; Keizer 1987):
TT:F: = IdcTe (dR=dTe )
2
RGef f (1 0 ) 1 + (! ef f )2
1
1
IF ;
(2.29)
where IF is the IF mismatch factor in equation 2.16. In addition to a ecting the thermal
uctuation noise, electro-thermal feedback also a ects the Johnson noise. In the absence of electro-thermal feedback, the Johnson output noise temperature would be equal to the average temperature of the electrons. However, the result is modi ed in the presence of electro-thermal feedback. The physical reason for this is as follows (Kogan 1996): Consider the case of a voltage bias at all frequencies. If the current
uctuates, then the power dissipated in the system
uctuates. Thus, the temperature
uctuates. But this causes a change in the resistance, which further changes the current under a voltage bias condition. Thus, temperature
uctuations and Johnson noise are coupled. The resultant expression for the Johnson output noise is given by (Mather 1982; Karasik and Elantev 1995):
Tout = Te
(1
0 )
1
2
1 + (! ef f )2
1 + (! th )2
!
IF :
(2.30)
This should be added to equation 2.29 to get the total output noise. Above it was argued that
would usually be much less than one when the load resistance of the ampli er was 50
, as is
typical for high frequency ampli ers. This fact will also cause the Johnson noise to tend toward the Johnson noise in the absence of electro-thermal feedback, since ef f
!
th when
! 0.
For the
remainder of this thesis, we will treat the Johnson noise as that in the absence of electro-thermal feedback because of this e ect.
2.1.4
Mixer noise
The mixer noise, de ned in equation 1.2, can now be calculated on the basis of the above calculations. The result is:
2 2Te G
Tmix (!IF ) =
PLO
+
1 + (!IF ef f )2 2 PLO Idc (dR=dTe )2
2RTe G2
:
(2.31)
26
The rst term is due to the thermal
uctuation noise, while the second term is due to the Johnson noise. There are a few important observations to make. First, the term due to thermal
uctuation noise is independent of the intermediate frequency, and it contains no e ects of electro-thermal feedback. Both of these are the result of the \
uctuation-dissipation" result: The
uctuations are related to the response to an external stimulus. Therefore, the spectrum of the
uctuations is the same as the frequency response to applied power. Additionally, the enhancement or suppression of the
uctuations at a given frequency due to electro-thermal feedback is the same as the enhancement or suppression of the response of the system to an external signal. The second term, due to Johnson noise, is dependent on the intermediate frequency. This is simply due to that fact that Johnson noise is white, whereas the conversion eciency decreases as the intermediate frequency is increased5 . As a result of this, the gain
bandwidth
(i.e., the intermediate
bandwidth,
frequency at which the conversion gain drops by 3 dB) is not necessarily equal to the noise which we de ne as the frequency at which the using equation 2.31, it is simple to show that:
mixer noise
increases by a factor of two. In fact,
fnoise3dB fgain3dB
s
=
TJohn: + TT:F: (0) : TJohn:
(2.32)
Typically, the thermal
uctuation noise if 50 K, and the Johnson output noise is 5 K, so that the noise bandwidth is predicted to be larger than the gain bandwidth by a factor of three.
2.2
Mixing in superconducting hot-electron bolometers: electron and phonon dynamics
In all of the above calculations, the thermal time constant was speci ed as C/G, and not investigated further. However, as was mentioned in the introductory chapter, this time constant is usually much too long to allow reasonable (i.e. multi-GHz) intermediate frequency operation. In this section, the dynamics which determine this time-constant in superconducting hot-electron bolometers will be discussed in more detail. In order to discuss these dynamics, we must also discuss the important
5 The Johnson noise is only white up to k T =h 100 GHz for T = 5 K , which is much higher than the frequencies e B e used in this thesis.
27
length scales of the problem. In the following, we will limit the discussion to the length and time scales in normal metals. Only later will the superconducting properties be discussed.
2.2.1
Electron-electron interaction rate
In dirty thin lms at cryogenic temperatures, the electron-electron interaction is enhanced compared to the electron-phonon interaction. If the rf power is being absorbed by electrons, the electronelectron interactions will be strong enough to force a steady-state electron energy distribution close to a Fermi-Dirac distribution. Therefore, an e ective electron temperature can be de ned. The electron-phonon interaction is weaker, yet will allow energy to
ow from the hot-electron system to the (cooler) phonon system. The Nb lms studied in this thesis have an elastic mean free path of 5
0 10 (Gershenzon, Gubandov, and Zhuravlev 1983). The inelastic electron-electron interaction A
0 ee1 (s01 ) 108Rsq T;
(2.33)
rate can be related to the resistivity, and for dirty lms is approximately given by
where T is the temperature in K and
Rsq
is the sheet resistance in ohms (Santhanam et al. 1987;
Altshuler and Aronov 1985). Thus, for a temperature of 5.5 K and a sheet resistance of 30
(typical of the thin lms studied in this thesis), ee
60
ps.
2.2.2
Electron-phonon interaction rate
The electron-phonon interaction rate in dirty metallic thin lms is dicult to calculate theoretically from rst principles. (For a review, the reader is referred to Mittal (1996).) Numerous experiments have measured the rate at various temperatures, and the experiments for di erent metals disagree.
01 In clean, bulk 3d metals, it is well established that the electron-phonon energy relaxation rate, e0ph ,
varies as
T3
at temperatures well below the Debye temperature. However, two separate e ects are
important in dirty thin lms which are expected to modify this behavior. First, disorder is expected to modify the electron-phonon interaction. Various theoretical calculations disagree about how this modi es the interaction rate. Second, in thin lms, the e ective dimensionality of the lm may be reduced from 3d to 2d. This can also change the power law. For the devices studied in this thesis,
28
then, a phenomenological approach must be taken to estimate the electron-phonon interaction rate. In this thesis, two separate methods are used to determine this: noise thermometry in the normal state and mixing measurements in the resistive state. The results obtained here agree quantitatively with the measurements of Gershenzon et al. (1990), who measured the electron-phonon interaction rate using mixing measurements and weak-localization measurements on a variety of Nb lms of di erent thickness and resistivity. In both Gershenzon et al. (1990) and here, the electron-phonon interaction rate is found to vary as with
T2
below 10 K. For the 100 lms measured in this thesis A
01 e0ph
Rsq
= 30
, both groups have found that the electron-phonon interaction rate at 4.2 K is (1
ns)01 .
2.2.3
Electron-phonon and electron-electron interaction lengths
In this subsection, we consider the transition from a macroscopic resistor to a microscopic resistor. Other than the length of the resistor itself, there are three other important length scales: the elastic mean free path, the electron-phonon length, and the electron-electron length. The electron-phonon length,
Le0ph
pD 0
m
e ph , is the length a hot-electron will di use before emitting a phonon, on
average. For the lms studied in this thesis, approximately 0:3
D
1
cm2 =s,
so the electron-phonon length is
at 4.2 K. The electron-electron length,
Le0e
pD 0 , is the length a
e e
hot-electron will di use, on average, before undergoing an inelastic scattering event with another electron. For the lms studied in this thesis, at 4.2 K,
Lee
0:09 m.
Lee .
The minimum length
scale over which is it meaningful to de ne an electron temperature is
For length scales smaller
than this, the electrons are not able to come to a steady-state local Fermi-Dirac distribution before di using away from the region of interest.
2.2.4
Thermal time constant: di usion and phonon cooling
In the previous section, the bolometer was treated as one lumped element, with a heat capacity C and a thermal conductance to the thermal \bath" G. However there are two mechanisms for the transfer of heat away from the electronic system: the electron-phonon interaction, and out-di usion of heat into the leads. The thermal circuit which indicates this is shown in gure 2.2. For the purposes of this thesis, the two thermal conductances are assumed to add. Thus, the thermal conductance to
29
DC RF power heats electrons
Diffusion cooling out ends
Electrons
Gdiff Ge-ph
Leads
Phonons
Gbd
Substrate
Figure 2.2: Thermal circuit diagram.
the bath G is modeled as two separate thermal conductances in parallel: boundary thermal conductance, conductance
G
=
Ge0ph + Gdif f .
The
Gbd ,
can be assumed to be much larger than the electron-phonon
Ge0ph
if the lm thickness is less than 10 nm (Gershenzon et al. 1990). Therefore,
in this approximation, the thermal time constant is the result of two separate, independent cooling mechanisms:
0 01 01 th1 = G=C = (Ge0ph + Gdif f )=C = Ge0ph =C + Gdif f =C = e0ph + dif f :
(2.34)
We now discuss the scaling of the thermal time constant with the bridge geometry, if we neglect
30
end e ects and if the thermal conductance of the substrate is high. The (electronic) heat capacity C is proportional to the volume of the electronic system. The electron-phonon thermal conductance
Ge0ph is proportional to the electronic volume also, since this is a bulk-e ect.
be independent of the bridge geometry.
Therefore, e0ph will
The cooling via out-di usion can be related to the bridge resistance via the Wiedemann-Franz law, and hence to its length. The Wiedemann-Franz law relates the electrical conductivity to the thermal conductivity in a metal6 , and is given by:
K thermal = LT electrical :
The prefactor will be worked out below, but it is clear that
(2.35)
Gelectrical
= 1=R. Thus,
the length, so that dif f
L . The net result is the sum of two terms, one of which is independent
2
Gdif f
L0 .
1
Gdif f
should be proportional to
Additionally, the heat capacity also scales linearly with
of the length, and the other depends on the length squared. Schematically,
0 th1 = A + BL02:
If the length is very long compared to
(2.36)
Le0ph , then the second term is negligible and the rst term
of the end, while the rest of the electrons are cooled by the
(the electron-phonon time) dominates. In this case, the di usion cooling is only useful for electrons that are within approximately
Le0ph
electron-phonon interaction, hence the electron-phonon interaction time is the relevant time for th . On the other hand, if the bridge is short compared to
Le0ph , then the second term dominates.
In
this case, a dramatic increase in the cooling rate is expected, which is desired in order to increase the intermediate frequency bandwidth. This is the central result of this section. In order to derive the crossover length quantitatively, it will be necessary to determine the constant of proportionally between the total electrical resistance and the total thermal conductance. To do this, it will be necessary to calculate the temperature pro le, which will be done in the following section. If the length of the bridge is less than
Le0e ,
then the concept of a local temperature within
the bridge cannot be meaningfully applied. The bridge is then in the \mesoscopic" regime, and a
6 At the temperatures used in this thesis (<10 K), the dominant source of scattering is elastic impurity scattering, so that the Wiedemann-Franz law is valid.
31
qualitatively new theory may be required for that case.
2.3
Temperature Pro le
Although the theory presented in the previous section predicts the conversion eciency and output noise of a lumped thermal element, the devices studied in this thesis should be treated as a distributed thermal system. For a distributed system, the output noise due to Johnson noise is predicted to be the average temperature along the length of the bridge. However, a quantitative theory for the conversion eciency and thermal
uctuation noise which treats the device as a distributed system has not yet been developed7 . Therefore, in this section we will calculate the temperature pro le under conditions of uniform dissipation of dc and ac power and attempt to relate the distributed system approach to the lumped element approach by deriving an e ective time constant and thermal conductance between the electrons and the bath. These quantities can then be used in the theory of section 2.1 as an approximation to expected device performance. The outline of this section is as follows: In the rst subsection, the temperature pro le for weak and strong dc heating will be discussed in the absence of any electron-phonon interactions. In the second and third subsections, the temperature pro le for weak ac heating will be derived, again in the absence of any electron-phonon interaction. In the fourth subsection, the results of numerical simulations in the presence of electron-phonon interactions will be presented. The most general case of dc and ac heating in the presence of electron-phonon interactions is discussed in the nal subsection.
2.3.1
Strong and weak dc heating
In the steady state, the
ow of heat and the electron temperature are governed by the timeindependent heat-di usion equation, namely
pin (x) 0 pout (x) = 0
d dT K e : dx dx
(2.37)
7 The case of a lumped element connected to a bath through a distributed system was considered in Mather (1982) and Karasik and Elantev (1995).
32
Here, pin (x) is the input power density delivered to the electrons, and pout (x) is the out
ow power density. The input power density is
Ptotal =V olume
=
I 2 R=V olume,
if the power dissipation is
uniform along the length of the bridge. The out
ow power density is zero if the electron-phonon coupling is neglected. As the coordinate system, take the bridge to lie between x=0 and x=L. The boundary condition is
T (x = 0; L) = Tb , the bath temperature.
This approximation assumes that
the conductivity of the metal leads is very large compared to the conductivity of the microbridge. It is useful to rewrite equation 2.37 in dimensionless variables, namely
T0 x0 p0
In this case, equation 2.37 becomes
T =T x=L R PLT R = ILT
e b total b
(2.38) (2.39)
2
2
2
b
2 =
LT
V
2
b
2
:
(2.40)
dT 0
dx0
2
0 d 2 T 0 = 0p0 : +T dx02
(2.41)
The solution to this equation for the electron temperature can be shown to be (Mittal 1995; de Jong 1995)
T0 Te (x)
= =
p1 + x0(1 0 x0)p0 ; or s x x PR T 1+ 10 : L L T L
b
(2.42) (2.43)
2
This result contains in principle all the information about the temperature pro le in the case of dc heating. For concreteness' sake, the pro le is illustrated in gure 2.3 for three di erent values of
p0 .
The temperature pro le is not directly measured in this thesis. To relate the above calculation more directly to measurements, we need to calculate the average temperature rise as a function of input power. The result is (Skalare 1994):
< Te >
=
Tb
2
rL + T
b
2
r P R s P R + arcsin 4T L + RP PR 4 L
1
b
2
(2.44)
33
2.0 p'=10 1.8
Telectron/Tbath
1.6
p'=5
1.4
1.2
p'=1
1.0 0.0
0.2
0.4 x/L
0.6
0.8
1.0
Figure 2.3: Temperature pro le in the absence of electron-phonon interaction, for various values of the input power, p0 P R=Tb2 .
L
Case:
Te (x) Tb
1 + x0 (1
< Te > Tb
1 + p0 =12
@P @ < Te >
Tmax Tb
1 + p0 =8
@P @Tmax
1Tmax < 1Te >
weak heating
0 x0 )p0=2
LTb R=12
R=
LTb R=8
R=8
b
1 :5
strong heating
px0 (1 0 x0 )p0
(=8)
pp0
LT
27 ( 2 )
b
(1=2)
pp0 LT
4=
1:27
Table 2.1: Limits of thermal pro le calculations in weak (P heating limits.
< LTb2 =R) and
strong (P
> LTb2 =R)
34
=
Tb
2 2 pp0 3 s p0 1+ p 0 + arcsin 2 2 4 + p0 p s r
1+
(2.45)
An additional result of interest is how hot the center gets, compared to the rest of the bridge. The maximum temperature of the bridge is given by
Tmax = Tb
PR 1+ 2 4LTb
= Tb
p0
4
:
(2.46)
Although the bolometer is really a distributed system, one can attempt to de ne an e ective lumpedelement thermal conduction to the bath due to di usion, in analogy with equation 2.35. There are two possible ways to de ne this, as (1) the average change in temperature with power, or as (2) the change in the temperature at the center with power. Both results can be related to the electrical resistance in the case of weak heating (p0
< 1) through:
= =
Gef f
=
LT =R
b
ef f ;
Ref f
R=12; def:(1)(P rober1993) R=8; def:(2)
(2.47) (2.48)
For strong heating, the 12 is replaced by
13, while the 8 remains unchanged. Table 2.1 summarizes
the results for limits of the above calculations for the weak and strong heating cases. A nal note is that the maximum temperature rise is only about 30-50% higher than the average temperature rise, in both the weak and strong heating cases.
2.3.2
Weak ac heating
When the source of heating is time-dependent, as in most of the experiments in this thesis, the equation that governs the temperature pro le is the time-dependent generalization of equation 2.37, namely:
@ @T (x; t) pin (x; t) 0 pout (x; t) = 0 K e @x @x
@ 0c T (x; t)1
+
@t
e
:
(2.49)
Here c is the speci c heat. In the case of weak heating, we neglect the dependence of the thermal conductivity and speci c heat on the (local) temperature. For this sub-section, we let the total bridge length be
L = 2l, and the bridge lie between x = 6l.
If the power dissipated in the device is
35
written in terms of its Fourrier components as
P (t) =
XP
!
w
ei!t ; Tb
and the ends (at
(2.50)
and if the initial conditions are that the device is initially at
x
=
6l) are
always at Tb , then the full solution to the time-dependent di usion equation can be calculated using Duhamel's theorem (Carslaw and Jaeger 1959). The author nds: 16l2
Te (x; t) = Tb +
where
2 K
1 XX (01)
+
! n=0
n
(2n + 1)
cos 3
"
(2n + 1)x 2l
#
P! i!t 0t= n e +e ; 1 + i! n
(2.51)
K
is the thermal conductivity, and
n
Here the di usion constant
L2 : 2 D(2n + 1)2
(2.52)
D is equal to the thermal conductivity K divided by the speci c heat c.
In the case where the heating is dc only, this reduces to the results of section 2.3.1.
The time-dependent temperature pro le is not measured in this thesis. However, an approximation to the average electron temperature over the length as a function of time is measured, so that is a quantity of interest. This can be calculated from equation 2.51 to be
< Te (x; t) >x= Tb +
+1 1 0 XX P! i!t 0t= n e +e : 2 C ! n=0 (2n + 1)4 1 + i! n
8
(2.53)
In the case being considered here, the power is never negative, so that there are always at least two terms in equation 2.50, one with
! = 0, and one with ! 6= 0, i.e. P (t) = P0 + Pw ei!t :
(2.54)
In that simple case, we have
< Te (x; t) >x= Tb + P0 12D C
L2
8 0 + ei!t P! 2 C
1 X
+
n=0
(2n + 1)4 1 + i! n
1
1
;
(2.55)
36
where the heat capacity
C is the speci c heat c times the volume.
The second term simply gives the
dc rise in the average temperature, and is equivalent to equation 2.47, if the Drude model expressions for c, D, and
L are used. In other words,
< Te >x=
P0 L2 C= 12D
=
P0 R LTb= 12 :
(2.56)
The third term in equation 2.55 is the ac component of the average temperature rise. Note that there is not one single time constant that characterizes this quantity. This is due to the distributed nature of the time-dependent heat
ow. Equation 2.55 is therefore not equivalent to equation 2.7, with a suitable rede nition of
.
However, it turns out that the rst term in the series in equation 2.55
dominates, and therefore, to a good approximation, equation 2.7 can still be used, provided an e ective time constant of
ef f
is used. This time constant is
not
=
L2 2 D
(2.57)
equal to the heat capacity divided by the dc thermal conductance
G, de ned in equation 2.47. In gure 2.4, the ac component of the average temperature vs. frequency is plotted for the exact result, equation 2.55, the approximation suggested in equation 2.57, and the (incorrect) result obtained by using the heat capacity divided by dc thermal conductance (eq. 2.47) as the e ective time constant. The conclusion of this subsection is then this: The distributed nature of the time-dependent heat
ow gives rise (approximately) to an e ective time-constant given by equation 2.57, which is
not
obtained by using the dc thermal conductance to calculate the time-
constant. An additional conclusion is that the di usion time constant is equal to e0ph when the bridge length is equal to behavior occurs at
Le0ph .
Therefore, the crossover from phonon-cooled to di usion-cooled
L = Le0ph .
2.3.3
Spatially distributed temperature
uctuations
In section 2.1.2, the
uctuations in the temperature of a single thermal element connected through a thermal conductance to the thermal bath was considered. A more appropriate model for the di usion-cooled bolometer is a distributed element approach. Each element is considered to be linked to its nearest neighbor through a thermal conductance. Fluctuations in the
ow between
37
1
9 8 7 6 5 4
τ=L2/12 D (incorrect)
/ ( Pω
x
L2/(12D C)
)
3
2
Exact result τ=L2/π2 D approximation τ=L2/12 D (incorrect)
τ=L2/π2 D approximation
0.1 0.1
2
3
4
5 6 7 8 9
1
2
3
4
5 6 7 8 9
10
ω (L2/π2D)
Figure 2.4: Distributed vs. lumped element temperature rise. The lumped element approximation 2.57 is close to the exact solution 2.55, but di erent from the dc lumped element approximation.
38
nearest neighbors are then postulated, and the magnitude of forcing function must be calculated. A generalization of the lumped element Langevin equation 2.18 can be derived (Landau and Lifshitz 1980), and is stated here:
@ (x; 0 @x K @Te@x t)
@ 0c T (x; t)1
+
@t
e
=
@f (x; t) : @x
(2.58)
Here the forcing function is random in both space and time, i.e.
< f (0; 0)f (x; t) >
(t)(x):
(2.59)
The solution to this equation is considered in van Vliet and Fassett (1965) and Voss and Clarke (1976). There, the case of an in nite one dimensional system is considered. For the devices studied in this thesis, the appropriate system to consider is a nite one dimensional system. The boundary conditions are that the temperature at the ends are xed. In equation 3.12 of Voss and Clarke (1976), the spectral density of the temperature
uctuations averaged along the length of a one dimensional system is expressed as an integral over a continuum of allowed k-vectors. If the ends of the system are kept at a xed temperature, then only discrete k-vectors are allowed. The integral can be converted to a sum, and the author nds the following for the resultant spectral density of the temperature
uctuations:
< Te (x; !) >x
2
=
2 4kB Te
0 2 C 8
1 X
+
n=0
(2n + 1)4 1 + (! n )2
1
1
:
(2.60)
This equation was derived when no external power is applied, so that Te is well de ned, and equal to the bath temperature. The conclusion of this section is that the spectral density of the temperature
uctuations is not equivalent to the lumped element case with a suitable rede nition of . The rst term in equation 2.60 again is the dominant term, and so the lumped element approximation can still be used to a very good approximation, provided an e ective time constant given by 2.57 is used. Finally, the low-frequency limit of equation 2.60 can be shown to be:
< Te (x; !) >x ! !0
lim
2
=
C= 12D
2 4kB Te
L2
=
LT =
2 4kB Te
R b 12
:
(2.61)
39
2.3.4
DC heating with electron-phonon interaction
So far, we have been neglecting the electron-phonon interaction. However, in the presence of electronphonon interaction, the di usion-equation 2.37 contains a \sink" term for the heat
ow: power can
ow from the electron system directly to the phonon system. The power
ow density depends on the electron temperature and the phonon temperature, as well as the electron mean-free-path. As discussed in section 2.2.2, there is no theoretical prediction that accounts for the strength or dependence of the electron-phonon coupling in Nb, so empirical results must be used. We state the result here, which was found in Gershenzon et al. (1990), as well as the experiments described later in this thesis. The electron-phonon coupling is given by:
4 pout = A(Te4 0 Tph );
(2.62)
where
A
= 1
0 2 10
10
W m03 K 04 for D e0ph = C Ge0ph
=
= 1
cm2 =s.
The electron-phonon time can be calculated
to give:
Te V dPe0ph =dTe
=
3 4ATe V
TeV
=
2 4ATe
:
(2.63)
In order to account for this quantitatively, the di usion equation 2.37 must be solved numerically. The di usion equation can again be cast in dimensionless variables, and the result is:
0
where
dT 0
dx0
2
2 0 0 d T + 0T 04 0 11 = p0 ; 0 T dx02
ATb2L2 L
= 1 4
(2.64)
L Le0ph
2
:
(2.65)
Before discussing this quantitatively, the general characteristics of the solutions can be predicted. For devices which are very long compared with near (i.e. within
Le0ph ,
the temperature pro le will be
at except
Le0ph
of) the ends, where di usion cooling will contribute somewhat. For devices
which are much shorter than
Le0ph ,
the electron-phonon interaction will not contribute at all, as
the electrons will di use out the ends before the electron-phonon interaction can contribute. Thus, the pro le will be given accurately by equation 2.43. For intermediate length devices, both e ects will be important. Note that
Le0ph
is a temperature-dependent quantity, so that devices that are
40
less than
Le0ph
for small heating may actually move into the intermediate limit if the temperature
in the center becomes signi cantly larger than the bath temperature. Since there is no analytical solution for equation 2.64, numerical methods must be used. The numerical code to solve equation 2.64 was written by Chalsani (1997). We plot in gure 2.5 several temperature pro les so generated, for bridges of length 6
Le0ph and 20 Le0ph .
These should be con-
trasted with gure 2.3. The pro les are in agreement with the discussion of the previous paragraph. In a later chapter, these simulations will be used to compare with experimentally measured values of
< Te >x as a function of input power, using noise-thermometry.
2.3.5
AC heating in the presence of electron phonon interaction; strong AC heating
Based on the above results, we can come to the following conclusions regarding the temperature pro le: For very long devices, in the presence of weak or strong dc or ac heating, the behavior should be that of a lumped element with a single time constant, e0ph . For devices much shorter than
Le0ph
in the presence of weak ac heating, a lumped element is a good approximation, with
a single time constant of
L2 =2 D.
A similar conclusion is expected to hold in the case of strong
ac heating, without electron-phonon interactions, which has not yet been calculated. A numerical calculation of the time-dependent di usion equation in the presence of electron-phonon interactions would be required to quantitatively evaluate the evolution of the behavior between the two regimes. However, we expect that the cooling rates should approximately add, and this approximation will be used in the remainder of this thesis.
2.4
Uniform vs. non-uniform dissipation of power
In the previous sections, the temperature pro le of hot-electrons in the device was calculated for dc and ac heating assuming that the local power density was constant. However, the resistance varies with the local temperature, so if there is a constant current
owing through the device, the local power density varies. In order to calculate the temperature pro le correctly as a function of input power, a self-consistent calculation would need to be done. The local temperature a ects
41
2.0
L/Leph=6
1.8 p'=50
Telectron/Tbath
p'=100
1.6
1.4 p'=10 1.2
1.0 0.0 2.0
0.2
0.4 x/L
0.6
0.8 p'=1000
1.0
L/Leph=20
1.8
p'=500
Telectron/Tbath
1.6
1.4
p'=100
1.2
1.0 0.0
0.2
0.4 x/L
0.6
0.8
1.0
Figure 2.5: Temperature pro le with electron-phonon interaction, for various values of the input power, p0 P R=T 2 , calculated by numerically solving equation 2.64, keeping R = 100
. The code for the simulation was written by Chalsani (1997).
L
42
the local resistance which in turn a ects the local power dissipation which feeds back to a ect the local temperature. However, such a calculation would be of dubious signi cance for the shortest devices measured in this thesis, since the local temperature is not a meaningful concept for length scales shorter than
Lee .
For this reason, the comparison of theory and experiment will neglect the
non-uniform dissipation of power. The uniform-dissipation case will be used as a guide, but not expected to hold quantitatively. A more sophisticated theory will eventually need to be developed if the detailed response to ac and dc power is to be predicted.
The impedance of the device at frequencies above the gap frequency however (
700 GHz in bulk
Nb) is constant and equal to the normal state impedance. Therefore, if a high frequency signal is applied above the gap frequency, then the dissipation of power
is
uniform. For the measurements
presented in this thesis, all frequencies used were below the gap frequency. Experiments done at JPL used ac signals above the gap frequency. The two sets of results will be compared in chapter 6.
2.5
DC and ac di erential impedance
The di erential impedance of the device is an important quantity to know for circuit design purposes. In addition, measurements of the di erential impedance can also test the underlying physical model. The simplest theoretical model available postulates that the di erential impedance at frequencies
0 0 well above th1 is simply Vdc =Idc . At frequencies below th1 , the electron temperature can follow the
(slow) change in dissipated power. However, at high frequencies the electron temperature cannot follow the fast power variation and so stays xed. Thus, it is possible to measure the di erential impedance of the electron system in the absence of self-heating. In this way, it is possible to measure the I-V curve if there were no power dissipated in the device. We call this underlying I-V curve the \isothermal" I-V curve. The isothermal I-V curve is expected to be simply a straight line with slope between 0 and
Rn , with slope determined solely by the temperature of the electrons.
Based on the heat
ow equation, the equivalent circuit shown in gure 2.6 can be derived (Elant'ev and Karasik 1989). This circuit diagram can be understood very easily. At low frequencies, the capacitance does not conduct current, and the di erential impedance is simply equal to the dc di erential impedance, the di erential impedance is
R1 + R2 .
This is
dV=dI .
At high frequencies, the capacitance shorts
R2 ,
so
R1 .
This is equal to
Vdc =Idc, as discussed above.
At high frequencies,
43
C R1
R2
Figure 2.6: Equivalent circuit of bolometer.
44
the time-varying power dissipated in the devices causes a time-varying temperature of the electrons which lags the power. Since the electron temperature is out of phase with the power dissipated, the resistance is out of phase with the power dissipated. This thermal inertia gives rise to the e ective capacitance in the model, whose value is set by the thermal time constant of the system. The parameters in the circuit model are:
R1 = Vdc =Idc; R1 + R2 = dV=dI dc; R2 C = th : 1 0 0
0
1
(2.66) (2.67) (2.68)
2.6
Voltage dependence of gain and noise; from I-V curve
When dc and ac power are applied to the device, the electron temperature is heated above the bath temperature to somewhere near the critical temperature. However, the temperature of the electron system is dicult to predict accurately. Since the value of dR/dT changes from 0 to its maximum value in the range of about 0.25 K, it is dicult to predict the value of dR/dT under operating conditions. This makes predictions of the conversion eciency and output noise based on equations 2.13 and 2.28 dicult. There is, however, a way to determine the value of
0
from
the measured I-V curve, which allows predictions of the output noise and eciency. The principle is simply this: An increase in bias voltage increases the power dissipated. This raises the electron temperature, which in turn causes an increase in resistance. Therefore, the quantity dR/dP can be measured. A derivation is given in Ekstrm et al. (1995) for the following formula: o
2 Idc dR=dP 2 = Idc dR=(GdTe ) =
(dV=dI )
(dV=dI ) + R
0R :
0
(2.69)
Therefore, the measured dc I-V curve can provide a measurement of calculate
0 .
(It is straightforward to
from the dc I-V curve once
0
is known.) The predictions of equations 2.13 and 2.28
can be rewritten in terms of
and 0
as
(!IF ) =
PLO 2 1 0 2 1 + (! 2 IF ; 2 Pdc (1 0 ) IF ef f )
(2.70)
45
Tout =
The values of
Te2 G 2 1 0 : Pdc (1 0 )2 1 + (!IF ef f )2 IF
(2.71)
Pdc; PLO ; G; and Te
can be estimated with reasonable accuracy, so that a prediction
of device performance from the measured I-V curve should be possible.
Chapter 3
Experimental technique
In this thesis, the conversion eciency and output noise were measured with rf and LO signals between 8 and 40 GHz. The goal was to measure the IF bandwidth and noise for devices of di erent lengths in order to determine if di usion cooling could be used to increase the bandwidth, and to determine whether di usion-cooled bolometers are competitive devices for THz heterodyne detection. Since the mixing process is thermal and depends only on heating of the electrons, the results of these measurements should be representative of device performance at THz frequencies. We expect that the bandwidth and output noise measured here should be very similar to results at 1 THz, but the detailed dependence on the dc and LO power may be di erent. These issues will be discussed in more detail in chapter 6, where the results presented in this thesis are compared with measurements on di usion-cooled bolometers done at JPL at THz frequencies. The availability of variable frequency sources at THz frequencies is very limited, so for a systematic study of device performance as a function of frequency and other parameters, a lower frequency was used. The experimental goal was to measure the conversion eciency and output noise of several devices as a function of dc and LO power, and intermediate frequency. The measurements were done as follows: Two coherent signals (the LO and rf) were applied to the device, and the frequency of the rf signal was varied. The absolute power coupled to the device at each rf frequency used was calibrated as discussed in section 3.5. The power generated at the intermediate frequency was ampli ed and measured on a spectrum analyzer. By carefully calibrating the gain of the ampli er chain, the 46
47
absolute power generated by the device at the intermediate frequency was determined. Then, the (coupled) conversion eciency (the ratio of the output power at the IF to the input power at the rf) was calculated. An important advantage of using coherent signals to measure the bandwidth and conversion eciency is the large signal to noise ratio and the fact that the measurements demonstrate de nitively that the device response is indeed heterodyne. An additional goal was to measure the output noise of the device as a function of temperature, dc and LO power, and intermediate frequency. For this, the noise generated by the device was ampli ed and measured on either a spectrum analyzer or on a broadband detector preceded by a band-pass lter. The noise of the ampli er itself was also ampli ed and detected, so this contribution had to be carefully calibrated and subtracted o in order to determine the contribution from the device itself. By knowing the output noise and the conversion eciency, it was then possible to calculate the
mixer noise,
which
is the important parameter for any detector. If the mixer noise is low enough, then the devices will be competitive THz mixers. In this chapter, the technique used to measure the dc and rf properties of the devices studied in this thesis is described. First, the rf setup is described. Next, the dc electronics are described. Next, the thermometry is described. Then, the shielding and ltering techniques used are described. Finally, the rf calibrations are described.
3.1
3.1.1
RF setup
RF block diagram
The experimental setup used in this thesis consisted of the device which was mounted on a variable temperature stage in a vacuum can, as well as several rf components which were immersed in a liquid He bath which surrounded the vacuum can. The setup was unique in that it allowed the simultaneous measurement of many di erent high frequency properties of the devices, including the high frequency di erential impedance, the high frequency output noise in both the normal and mixed states, and the conversion eciency as a mixer as a function of dc power, LO power, LO and rf frequency, and temperature. A block diagram of the rf measurement system is shown in gure 3.1.
48
The rf and LO signals were coupled in weakly through a directional coupler1 . The attenuators were in place to reduce blackbody radiation from room temperature, as well as to damp any frequency dependence of coupling to the device, i.e. standing waves. The attenuator in front of the ampli er was removed during most runs where the noise was measured. The cooled low-noise ampli er2 was used to amplify any coherent signals generated by the device at the intermediate frequency, as well as noise generated by the device. A hermetic feedthrough SMA connector3 allowed dc and ac signals to be coupled to the device, which is mounted on a variable temperature stage inside a vacuum can. A 1" section of stainless steel (both inner and outer conductor) coax provided a weak thermal link between the He bath and the device mount, so that the temperature of the device mount could be independently varied from 1.8-20 K. (The rf loss of the stainless dc bias tee and hermetic connector combined was measured to be less than 1 dB below 10 GHz and less than 2.3 dB from 10-20 GHz at 4 K. The loss of the 1" section of stainless steel coax is less than 0.3 dB below 20 GHz.) By measuring the ratio of the intermediate frequency power generated by the device to the rf power delivered to the device, the (coupled) conversion eciency could be measured. In addition, the ampli ed output noise generated by the device could be measured, and with a proper calibration, the absolute noise generated by the device could be measured. An isolator was sometimes placed between the device and the ampli er to reduce the e ect of the change of impedance on the calibration coecients. This will be described in more detail in a chapter 5. Finally, the directional coupler allowed the device impedance to be measured: A weak signal sent in via the directional coupler will be re
ected o of the device and ampli ed, then measured at room temperature on the spectrum analyzer. By biasing the device in the superconducting state, the impedance is essentially zero, so that all the power is re
ected. This provides a (scalar) calibration of the coupling and gain, and allowed the (scalar) coupling to the 50
system to be measured when the device is in the normal or intermediate state. This is important for measuring the conversion eciency, as well as the noise, since the coupling factor must be included in the corrections to the
1 TRM (Manchester, New Hampshire) model DCS-210. For later experiments, a broader-band (1-18 GHz) coupler was used, Mac Technology (Klamath Falls, OR) model CD 4238-20F. 2 Miteq (Happauge, NY) model AFS3 00100800-32-CR-4. The ampli er was modi ed by the factory to be cryogenically coolable, and biased at 3 V to reduce power dissipation, instead of the recommended 6 V. The power dissipation in the ampli er was about 180 mW. 3 M/A-COM (Waltham, MA) model 2098-3251-94, soldered into place with Sn/Pb solder.
He Transfer Dewar
IF Path Out
Low Noise Amp (0.1-8 GHz) TN = 25 K 3 dB attenuator 10 dB attenuator
Spectrum Analyzer (HP 8593E) 9 kHz - 22 GHz
20 dB Directional Coupler
RF/LO Path In
DC Bias In SS Coax (1") Indium-sealed vacuum can
Bias-Tee
Mixer Block (1.8 - 20 K) 1.8 K He
49
Figure 3.1: Block diagram of RF measurement system. All components are connected with SMA connectors.
50
data.
3.1.2
Mixer mount
The mount used to couple from the coax system to the device is shown in gures 3.2 and 3.3, and the lead geometry of the chip is shown in gure 3.4. The coax transmission line was fed into a microstrip transmission line, both having a characteristic impedance of 50
. The microstrip was constructed of 0.025" thick copper plated te
on4 . One side of the te
on was etched by a commercial circuit board company5 to form the microstrip, and the other side was left plated. The Duroid was soldered onto the Cu block using low temperature solder. At the end of the microstrip, the device was \
ip-chipped" into place: The chip was carefully placed upside down on the end of the microstrip with indium squashed between the chip and the microstrip to provide reliable electrical contact. A spring-loaded pogo pin applied pressure to keep the chip in place. This mounting technique was extremely robust. After each device was mounted, the whole block was dipped into liquid He and warmed to room temperature three times, while the I-V curve was monitored to ensure the electrical contact did not change. All mounted devices survived this ordeal, and were then carefully handled when being inserted in the experimental setup. Once a device was mounted at the end of the microstrip, it could be inserted and removed from the experimental apparatus simply by screwing it onto the SMA connector of the apparatus. The coupling from the SMA connector of the mount to the device was measured by measuring the return loss of the device in the normal state, and found to be better than 90% between 50 MHz and 12 GHz. Additionally, a measurement of a chip resistor similarly mounted at room temperature showed a 90% or better coupling from 50 MHz to 40 GHz at room temperature. Thus, this method provides a
broadband, resonance free coupling
to a 50
device from 50 MHz to over 40 GHz. Most
mounting methods used to study mesoscopic electronic devices at cryogenic temperatures include some parasitic capacitance and inductance. The development of an ultra-broadband coupling scheme for the hot-electron bolometers was therefore a signi cant accomplishment.
4 5
Rogers Corporation (Chandler, Arizona) model RT/Duroid 6010LM, = 10:2. Tech Circuits (Wallingford, CT).
SMA Connector/ Microstrip Launcher 0.01" Quartz 50 Ω microstrip Substrate (Cu on duroid) Spring Loaded Pogo Pin Indium Blobs
0.025"
Figure 3.2: Mixer mount, side view (not
Cu Block (cylindrical) (rectilinear)
to scale).
Nb Microbridge w/ Au pads
51
52
(cylindrical)
SMA Connector/ Microstrip Launcher
(rectilinear) 50 Ω microstrip (Cu on duroid) Duroid (teflon)
Cu pedestal Chip with device placed upside down at end of microstrip Cu Block
Figure 3.3: Mixer mount, top view (to scale, 5:1).
53
1120 µm
100 µm
326 µm
Quartz substrate 3000 Å Au leads
Figure 3.4: Lead geometry, to scale 100:1. (Electron beam lithography is used to de ne the device length inside of the 6 m gap.)
1520 µm
6 µm gap for device
54
3.2
DC electronics
The circuit diagram for the dc electronics is shown in gure 3.5. This is a versatile bias scheme. The voltage can be controlled externally by computer or through the internal oscillator and dc bias. The 10
resistor in parallel with the device in combination with the 10
resistor in series with the device gives rise to a load line of 20
. The I and V outputs on the right are actually passed back through the rf lters, and through the adder box before being sent to the DVM or oscilloscope, which displays the I-V curve in real time. To switch to current bias, the 10
resistor in parallel with the device can be disconnected, and
Rbias
is used as the current sense resistor with the rst AD624
instrumentation ampli er. The connection from the bias box to the dc bias tee was all through semi-rigid coax with SMA connectors in order to avoid rf pickup. Additionally, a 2 MHz low pass lter6 was placed at the output of the bias box. The readout circuit is a two-point measurement of the resistance. This was not a problem, since the the lead resistance was low, less than 1
. This was measured by measuring the resistance when the device was biased in the superconducting state. Of particular interest is the way the devices are connected and disconnected from the electronics to avoid damage due to electrostatic discharge. The procedure was as follows: Before connecting the dc bias line to the bias box, all electronics in the bias box were powered up. In the initial con guration, switches S1 were and S3 are closed, and switch S2 is open. (Switch S2 serves to connect or disconnect the device from all of the electronics inside the bias box.) After the device line is connected, switch S2 is closed, connecting the device to the electronics. Next, switch S1 (which so far insured the ampli er leads would not
oat to a non-zero potential di erence) is opened. Finally, switch S3 is opened and the 10 k
potentiometer is slowly opened7 , allowing current from the electronics to slowly begin
owing through the device. In later experiments, switches S1 and S2 were rarely used, but switch S3 and the potentiometer were a sure way to ensure that when the coax was connected to the bias box, the center conductor would be at zero voltage. Only one of the six devices measured was damaged during the course of over fty separate experiments that were performed on the devices described in this thesis.
6 7
Mini-Circuits (Brooklyn, New York) model BLP-1.9 Since the device resistance was only 50-100
, the 10 k
is essentially an open circuit and draws little current.
ADDER BOX
+ 10 nF INA105 1 MΩ
RF π filter between boxes 700 kHz low pass
BIAS BOX
G=1 buffer
Vin1
10 kΩ
feedback circuit on/off
Vin2
1.5 kHz low pass filter
10 MΩ 1 MΩ 100 kΩ
Σ
Rbias (1 or 10 kΩ)
G=1,100, or 1000 + 1.5 kHz low pass filter
10 kΩ 10 nF 1 nF OP07 +
G=1 differential buffer
10 Ω
100 kΩ
S1
AD624
-
I+ I-
10 Ω
0.01-1 kHz 10 mV - 10 V
V+
1 MΩ (V- grounded)
Coarse, fine dc bias
1 MΩ
G=1,100, or 1000 +
1.5 kHz low pass filter
G=1 differential buffer
S2
S1
AD624
-
10 kΩ 10 nF
V+ V-
S3
10 kΩ
Figure 3.5: DC electronics schematic.
Device
55
56
3.3
Cryogenic setup and thermometry
The temperature of the block the device was mounted on was monitored by a calibrated diode thermometer8 whose case was screwed directly to the back of the Cu block. The temperature of the block was raised above the He bath temperature by applying current to a 1 k
resistor which was thermally anchored to the block. A commercially available feedback circuit9 under computer control via a GPIB interface adjusted the heater power to regulate the temperature to a precision of about 10 mK. The thermal time constant for the entire Cu block was about 30 seconds. Two consistency checks ensured that the thermometry was correct. experiments, the In a separate series of
Tc
of a Nb lm fabricated by A.A. Verheijen with
Tc
of 8.6 K was measured in
two separate setups. In the rst setup, the mount described in gure 3.2 was used with the diode thermometer. This setup was used both with and without a coax connected in order to verify that the heat load of the center pin was not heating up the device. In the second setup, a di erent mount designed for dc measurements only was used, together with a di erent, carbon-glass calibrated resistance thermometer10 . Both measurements of
Tc
agreed to within the speci ed accuracy of the
diode thermometer of 50 mK. A second consistency check comes from measuring the Johnson noise as a function of indicated temperature, and will be discussed in section 3.5. There, the thermometry and the microwave ampli er gain calibrations are shown to be self-consistent.
3.4
Shielding and ltering
Shielding and ltering were required for two reasons: rst, to avoid external heating of the device by spurious signals, and second to avoid confusing spurious signals from signals generated by the device. In this section, the shielding setup is rst described, and then estimates and measurements of the e ectiveness of these techniques are described. The technique used was to ensure that all connections to the device were done through semirigid coax cables with SMA connectors. While the isolation of the cable itself is typically 130 dB or greater at rf frequencies, the SMA connectors used have speci ed isolation of at least 80 dB from
8 9 10
Lakeshore model DT-470-BR-13 1.4L. Lakeshore model DRC-93 temperature controller. TRI Research model CC1000-5%.
57
dc to 18 GHz. On the dc bias out of the bias box, a 2 MHz low pass lter with loss greater than 40 dB from 5 MHz to 10 GHz was used. Therefore, the electromagnetic coupling to the device at all frequencies from dc to rf was well controlled. A similar scheme was used with the dc bias line for the cooled ampli er, but the last inch or so was twisted pair wiring due to the con guration of the commercial ampli er. Experimentally, it was possible to estimate stray coupling from coherent signals such as cell phones and radio and TV stations as follows: Some fraction of the power coupled into the device would get re
ected and fed into the input of the ampli er, whose gain was accurately characterized as described in section 3.5. Using this technique, it was determined experimentally that the largest absolute coherent signal coupled into the ampli er above about 100 MHz was due to cell phone signals at 900 MHz, with an absolute magnitude of less than 10015 W. The power coupled to the device at 900 MHz was most likely of the same order of magnitude or less. This power will not provide any signi cant heating of the electrons since the thermal conductance to the bath is of order nW/K. Additionally, the coherent signals generated by the device during the experiments described in this thesis were between 0.1 pW and 100 nW, so that the low-level spurious signals were not important. Stray coupling of incoherent signals such as 300 K black body and noise generated by the microwave oscillators was much more dicult to measure, and for this purpose room temperature and cooled attenuators were used wherever possible. The attenuation between the rf coax connector at the cryostat to the device was approximately 30 dB, so that the 300 K blackbody propagating down the coax would be attenuated to 0.3 K by the time it reached the device. Extraneous noise sources above 300 K from the microwave sources were carefully measured and attenuated down to below 300 K equivalent noise temperature.
3.5
Calibrations
Two separate techniques were used to calibrate the rf power coupled to the device, as well as the gain and noise of the ampli er, the rst using coherent signals from a microwave generator as the source of power, and the second using Johnson noise generated by the device as an incoherent source of known spectral intensity. For the rst set of experiments described in this thesis where only the
58
conversion eciency was measured, the calibrations with a coherent signal were sucient. However, for the second set of experiments where both the noise and the conversion eciency were measured, it was necessary to use the calibration coecients determined using the Johnson noise of the device.
3.5.1
Calibrations using a coherent signal
In one technique, a coherent signal was used as a source, and the ratio of the input power at a xed frequency to the output power at a xed frequency was measured in order to determine the ampli er gain and the rf coupling to the device. Using this technique, it was possible to calibrate the coupling from the rf input connector of the cryostat to the connector on the bias tee which fed directly through the hermetic feedthrough and 1" section of SS coax to the device mount. The loss of the bias tee, the hermetic SMA feedthrough, the 1" section of SS coax, and the on-chip microstrip was neglected in these measurements. The coherent calibration method required several sets of measurements of cold components and cables, which were then used as the coupling coecients for the rest of the experiments. A disadvantage of this technique is that it cannot be used to measure the noise of the ampli er, and it is does not account for run to run variation in the performance of the system. In a separate experiment, the drift of the calibration coecients as a function of time during the run was measured to be less than 1 dB over the course of the run. Due to this and other errors, the estimated error on the measured conversion eciency is 2 dB. The calibration coecients determined using a coherent source are plotted in gure 3.6. The top graph displays the loss between the connector at the entrance to the cryostat and the cooled directional coupler, as well as the coupling through the directional coupler to the device port. The rf power coupled to the device was measured by measuring the power generated by the source on the spectrum analyzer at each frequency, and then subtracting the coax cable loss and coupling of the directional coupler. The lower graph shows the gain of the ampli er chain used.
3.5.2
Calibrations using incoherent signals
and
In order to calibrate the ampli er gain
noise, the device itself in the normal state was used as
a source of known spectral intensity. If there is no current
owing through the device, then the
uctuation-dissipation theorem guarantees that the spectral intensity of the power radiated by the
0 Coax cable loss -10
Coupling (dB)
59
-20 Directional coupler
-30
-40
-50
0
2
4
6
8 10 12 Frequency (GHz)
14
16
18
20
50
40
System Gain (dB)
30
20
10
0
0
2
4
6 8 10 Frequency (GHz)
12
14
Figure 3.6: Calibrations coecients determined using a coherent source.
60
0.6
Noise power (1.25-1.75 GHz, arb. units)
Measured data linear fit 0.5 Tnoise = 22.609 ± 0.09 K Gain = 101.99 ± 0.22 K-1 0.4 0.3 0.2 0.1 0.0 -30 Tnoise
-20
-10
0 10 20 Block temperature (K)
30
40
Figure 3.7: Calibration at xed frequency of ampli er gain and noise.
device into a matched load is simply
kB T ,
where T is the temperature of the device. By varying
the temperature of the device and measuring the output power of the ampli er, it is possible to simultaneously determine the gain and noise of the entire ampli er system referred to the terminals of the device. In gure 3.7, the measured ampli ed noise power in the 1.25-1.75 GHz band (in units of volts on the detector) is plotted against the mixer block temperature, which is equal to the device temperature since there is no current
owing in the device for this experiment. For this plot, the data were taken at two minute intervals in order to assure that the mixer block temperature had stabilized before taking the next point, for a total sweep time of about 25 minutes. The line would intercept zero if the ampli er were noiseless; the non-zero intercept is the ampli er noise temperature, and the slope is the gain of the ampli er system. This technique assumes that the device is well-matched to the 50
input impedance of the ampli er, which is indeed the case as was measured in separate series of experiments to be discussed later. The technique of measuring the ampli er noise and gain can be applied at all frequencies, and
61
this was done in many of the experiments described below. Since the statistical variation in the noise power measured during a time
t in a bandwidth of B is proportional to 1= Bt, it is better to
p
integrate as long as possible, and to use as wide of a bandwidth as possible. The practical constraint is that the liquid He only lasts a few hours, and the gain and noise drift on a slow timescale. The compromise used when measuring the ampli er gain and noise as a function of frequency was as follows: The spectrum analyzer was set to a resolution bandwidth of 3 MHz11 , and an integration time per point of 0:16 s. 1000 traces were averaged, giving an e ective integration time of 0:16 ms, and then the data were smoothed over a 100 MHz bin. Therefore, the measured gain and noise were actually the average gain and noise over a 100 MHz bin, this poor spectral resolution being a disadvantage of the technique over using a coherent signal. This is especially a disadvantage when trying to measure the gain of the ampli er at frequencies below a few hundred MHz, where the gain is dropping rapidly to zero for the ampli ers used in these experiments. The gain of the ampli er system determined using the two separate techniques described above is plotted against frequency in gure 3.8. The disagreement (averaging 3 dB from 0.1 to 6 GHz) is probably due to two factors: First, the spectrum analyzer used has a rated accuracy of (1 nW), and second, there is
1 dB of loss between the device and the input of the ampli er, which
6 1 dB at the powers levels measured
is not accounted for in the coherent calibration. (The ampli er gain from the incoherent calibration was consistent within 0.5 dB over four runs on three separate devices.) An additional calibration of the microwave ampli er at a xed frequency was done with a (macroscopic) chip resistor, and the results obtained there were consistent with the results obtained when using the device itself as a source of incoherent power of known spectral intensity. Since the measurement of the device noise relies only on the thermometry, the incoherent calibrations were used for determining the device output noise. However, the absolute gain determined using a coherent source was more reliable for measuring the absolute conversion eciency of the device, so the gain determined using a coherent signal was used in that case. The gain of the ampli er system when con gured to use the Johnson noise of the device to calibrate the gain and noise is much higher than that plotted in gure 3.6, in order to insure that the ampli ed Johnson noise overcomes the input noise (
100; 000 K ) of the spectrum analyzer
11 This was the widest available resolution bandwidth. If enough rf equipment were available, a better and more ecient technique would have been to construct a custom spectrum analyzer with a much wider resolution bandwidth.
62
74 72 70
System gain (dB)
68 66 64 62 60 58 56 0 1 2 3 4 5 Frequency (GHz) 6 7 gain using noise as source gain using coherent source
Figure 3.8: Calibration of ampli er gain using coherent vs. incoherent source.
63
used12 . The nal ampli er has a 1 dB saturation point of typically 1 mW, so the gain must be less than 85 dB to ensure that the ampli ed noise of the rst ampli er ( does not saturate the last ampli er. It was also possible to measure
in-situ
25 K over a 10 GHz band)
the the absolute power coupled into the device using a
coherent source. This was done using a combination of noise-thermometry and the method of dc substitution as follows: First, the ampli er gain and noise was calibrated as discussed above. Next, dc power was applied to the device to heat the electrons while it was in the normal state. The heated electrons have a higher Johnson noise, and the rise in electron temperature with applied dc power was measured. Finally, the dc power was turned o and ac power was applied. Figure 3.9 shows the rise in electron temperature as a function of dc power and ac power calibrated using the (separate) technique of coherent signals discussed above. (The discontinuous change in the response to the ac power is due to a range-change on the microwave generator, which was not accounted for in the corrections.) This gure shows that the two methods for calculating the ac power agree fairly well. Therefore, it was later
assumed
that they agree, and the method was used to calculate how much
power was coupled into the device as a function of frequency while in the normal state, in absolute units. The advantage of this technique for calibrating the rf power coupled to the device is that it is
in-situ,
accounting for all loss up to the terminals of the device, and that it accounts for run to
run variations in calibration coecients.
3.5.3
Drift of calibrations during run
The drift of the gain and noise temperature of the ampli er during the course of the run was measured during a separate run in order to determine its magnitude. A three-temperature (incoherent) calibration was done every 10 minutes to monitor the gain and noise, using the spectrum analyzer and a detector preceded by a 1.25-1.75 GHz band-pass lter. The percent variations vs. time are plotted in gure 3.10, and are shown to be approximately 5%. The reason for the drift could be due to many factors which have not been analyzed in detail. Additionally, it is clear that the spectrum analyzer technique drifts more than using a direct detector, presumably because the power detection circuits within the spectrum analyzer itself are not as stable as the direct detector used.
12
Hewlett-Packard model HP8593E.
64
9.0 DC power AC power (8.5 GHz) 8.5
Electron temperature (K)
8.0 7.5 7.0 6.5 6.0
0
20
40 60 Power dissipated (nW)
80
100
Figure 3.9: Electron temperature vs. applied power, dc and ac.
65
10
Percent variation, using crystal detector
100 x (Tn-)/ 100 x (Gain-)/ 5
0
-5
-10 10
0
1
2
3 Runtime (hours)
4
5
Percent variation, using spectrum analyzer
100 x (Tn-)/ 100 x (Gain-)/ 5
0
-5
-10
0
1
2
3 Runtime (hours)
4
5
Figure 3.10: Drift of gain and noise with time in the 1.25-1.75 GHz band.
Chapter 4
Device geometries and dc properties
In this chapter, the geometry and dc properties of the devices studied in this thesis are described. The geometry and fabrication are discussed in the rst section. In the second section, measurements of resistance vs. temperature are discussed. In the third section, the general characteristics of the I-V curves are discussed.
4.1
Fabrication and geometry
The devices studied in this thesis were fabricated at the Center for Space Microelectronics Technology, Jet Propulsion Laboratory (JPL) by Bruce Bumble and H. G. LeDuc. Devices were fabricated for the tests described in this thesis, as well as the 500 GHz, 1.2 THz, and 2.5 THz tests in the research group of W.R. McGrath at JPL. The goal was to test devices from the same fabrication batch at JPL and Yale at di erent frequencies in order to allow quantitative comparison of results. In practice, devices from di erent batches were measured1 , and the comparisons will be discussed in chapter 6. Here, a brief summary of the fabrication details are given, since they are important to the
1
For this thesis, devices from fabrication batch 08/31/95 were used.
66
67
interpretation of the results. For more detail, the reader may consult Bumble and LeDuc (1997). The nished device consisted of a 10 nm thick Nb bridge which was connected to thick, wide gold leads. The 10 nm Nb lm extended under the entire gold lead, and this was made possible by a self-aligning fabrication process developed by B. Bumble. The Nb under the thick gold is assumed to be in the normal state due to the proximity e ect, so that the leads consist only of normal metals. In the rst step, a uniform 10 nm Nb lm is magnetron sputtered, followed
in-situ
by a uniform
coating of 15 nm of gold. This ensured that no oxide layer forms on the Nb lm, and that the contact resistance to the Nb is small. Next, electron-beam lithography is used to pattern a 25 nm thick gold line which is later used as an etch mask to de ne the width of the Nb bridge. Next, a 150 nm thick gold etch mask which de nes the length of the Nb bridge is patterned, again using electron beam lithography. A three-step reactive ion etch process is then used. In the rst etch, the gold over the Nb lm and some of the gold over the bridge is etched. In the second etch, the Nb lm everywhere but the bridge is etched using a process that selectively etches Nb. In the third etch, after the leads are deposited and patterned using optical lithography, the remaining gold on the bridge (as well as some from the leads) is etched, leaving the desired geometry. An SEM of a nished bridge is shown in gure 4.1. The length and width of the devices measured in this thesis were determined by inspecting the SEM image of di erent devices with the same design length in the same fabrication run. The estimated error using this technique is approximately
60:05 m. The
devices measured in this thesis were not measured in an SEM, in order to avoid electrical damage. The device geometries and dc resistances are documented in table 4.1. In order for di usion cooling to be ecient, it is important that the out-
ow of heat into the thick gold leads be unimpeded. By biasing the device in the superconducting state, it was possible to estimate the contact resistance. The contact resistance was less than 1
for all the devices measured, which is small enough compared to the bridge resistance to be neglected for the purposes of the work described in this thesis. The lengths fabricated range from well above Leph to well below Leph in order to allow the crossover from phonon dominated to di usion dominated behavior to be investigated. The length to width ratio was designed to achieve the same normal state resistance for each device, so that electro-thermal feedback e ects would be the same for each device, and only the length would a ect the time constant. As can be seen from the table, the resistances are all within about 50% of one another. The sheet
68
1000 Å Au Width Microbridge bolometer 100 Å Nb Length
1000 Å
Figure 4.1: SEM of a Nb microbridge with gold pads.
resistance is determined using device E, where the relative errors in the geometry are small, and is approximately 29
, corresponding to a resistivity of 29
-cm.
For metals, the normal state conductivity is related to the di usion constant by the Einstein relationship, the constant of proportionality depending on the Fermi velocity. It is important to know the di usion constant, since this determines the di usion cooling rate. In Gershenzon et al. (1990), the constant of proportionality between the conductivity and the di usion constant was determined for 10 nm Nb lms by measuring the conductivity from the geometry and the di usion constant from the slope of
dHc2 =dT
near H=0, which gives the di usion constant from a standard
result of the BCS theory of superconductivity. For the resistivity of the devices measured in this thesis, using the results of Gershenzon et al. (1990), D
1
cm2 =s.
4.2
dR=dTe
Resistance vs. Temperature
as well as
The resistance as a function of temperature was measured for the devices in order to determine
Tc.
These are plotted in gure 4.2. For a bulk superconducting sample, the
transition width should be very narrow, but for the samples studied here, the transition width is
69
Device: A1 A2 B C D E
Length (m) 0.08 0.08 0.16 0.24 0.6 3
Width (m) 0.08 0.08 0.08 0.08 0.2 1
RN (
)
56 56 80 96 93 86
dR/dT (
/K) 140 200 250 -
Table 4.1: Device geometries and dc resistances.
approximately 0.1 to 0.5 K. So far, no microscopic model which explains the nite transition width is available. There are two possible sources for the nite width. First, the critical temperature may vary spatially along the length of the bridge. Second, regions of the bridge may be
uctuating in and out of the superconducting state, at a rate which depends on the temperature. This second source of broadening might also contribute excess noise, but with a di erent time scale than dif f . Thus, our noise measurements tend to rule out this second explanation of the broadening.
4.3
Current-Voltage Characteristics
Figure 4.3 shows a typical dc I-V curve for a di usion-cooled device measured at a bath temperature of 2 K. For small bias voltages, there is no dissipation since the resistance is zero. At very large bias voltages, the dc power being dissipated heats most of the electrons signi cantly above
Tc.
If
the region near the ends is still superconducting, there will always be a little bit of excess current (compared to the normal state I-V curve), and this is indeed found to be the case. Finally, the region near the \dropback" is where the electrons are near Tc , and the I-V curve is no longer linear. This non-linearity is presumably due at least in part to dc heating, and is the e ect which allows the bridge to be used as a detector. While these general characteristics are understood, there is as yet no theory which allows quantitative predictions of the shape of the I-V curves. By applying ac power, it is possible to drive the whole bridge into the normal state, so that the I-V curve becomes
70
100
80
Resistance (Ω)
Device C L=0.24 µm 60 40 Device B L=0.16 µm Device A1 L=0.08 µm 20
0 4.0
4.5
5.0
5.5 6.0 Temperature (K)
6.5
7.0
Figure 4.2: Resistance vs. temperature curves for di usion-cooled devices.
71
100
50
Current (µA)
0
-50
-100 -3
-2
-1
0 Voltage (mV)
1
2
3
Figure 4.3: I-V curve for device A1, no ac power applied, for bath temperature of 2 K. The slight asymmetry is due to the sweep direction; points of unstable bias between the origin and 0.5 mV are not plotted. The slight slope of the superconducting branch is due to the lead and contact resistance. a straight line with slope
RN .
Chapter 5
Measurement of gain, noise, and bandwidth
This chapter contains most of the experimental results presented in this thesis. The measurements of the noise and conversion eciency presented in the rst section demonstrate that di usion cooling can be used to increase the intermediate frequency bandwidth to larger than 6 GHz, which is large enough for the use of hot-electron bolometers in THz receivers. The measurements of the output and mixer noise also indicate that the device is very sensitive, and indeed indicates the hot-electron bolometers will be the most sensitive devices for THz frequencies. In addition to measuring the performance as a practical device, the high-frequency impedance of the devices was also measured in order to test the physical model presented in chapter 2. These measurements are presented in the second major section, and indicate that the devices were all well coupled to the 50
rf system, in accordance with theoretical predictions. In a separate, but related set of experiments, noise thermometry was used to independently determine the response of the devices to ac and dc power when in the normal state. This allowed the crossover from di usion-cooled to phonon-cooled behavior to be determined in a separate set of experiments, thus substantiating the conclusions of the rst section. In the nal section of the chapter, the device output noise and conversion eciency are compared to theoretical predictions based on chapter 2. It is important to understand the device performance in order to predict the optimum mixer noise that can be attained, and how to obtain it. 72
73
While there is qualitative agreement between the basic theoretical predictions and the experimental results, a theory which quantitatively predicts device performance under a wide variety of operating conditions is still lacking.
5.1
Gain and noise
The measured gain, output noise, and mixer noise all depend on several parameters under experimental control for a given device, including the LO power, dc power/voltage, temperature, and intermediate frequency. Therefore, this section is divided into four subsections, in order to illustrate the dependence of the gain and noise on the di erent parameters. In the rst subsection, the variation of the dc I-V curve itself with temperature and LO power will be presented. Next, the dependence of the gain, output noise, and mixer noise on the LO power and dc voltage is presented. Third, the dependence of the gain on intermediate frequency is presented, together with the dependence of the bandwidth on device length. In the fourth subsection, the dependence of the output noise on intermediate frequency is presented. In the nal subsection, the dependence of the noise on the LO frequency is presented.
5.1.1
I-V curves vs. bath temperature, LO power
In gure 5.1, I-V curves are plotted for device A1 under several di erent bath temperatures. Very near
Tc, there is no hysteresis, although there is negative di erential resistance.
As the bath tem-
perature is lowered, a supercurrent branch develops. There, the power dissipated is zero since the voltage is zero. (The nite slope is due to resistance in the on-chip as well as o -chip leads.) Since there is no dissipation of power, the electronic temperature is equal to the bath temperature. The amount of current that can
ow without a voltage drop is not in nite. Therefore, there is a branch of the I-V curve where the power dissipated is enough to heat the electrons substantially above the bath temperature, hence bringing them (mostly) into the normal state. We have termed the nite-voltage region of the I-V curve near the point of switching the \dropback" region. It is here that just enough power is being dissipated to heat the electrons to near
Tc.
There is a simple theoretical model which predicts the shape of the I-V curves more quantitatively,
74
but the agreement with the experimental data in this thesis is marginal. The curve can be predicted as follows: assume that the electrons are a given temperature near or above Tc , and then infer the dc resistance from that temperature based on the resistance vs. temperature curve measured separately with small enough current to avoid self-heating. Given the assumed temperature, calculate the dc power necessary to heat the electrons above the bath temperature using knowledge of the thermal conductance. Then, from the dc power and resistance, infer the voltage by equating The current will simply be
Pdc
to
V 2 =R.
V=R.
This method requires quantitative knowledge of the amount of
power required to heat the electrons above the bath temperature, as well as a knowledge of the temperature pro le along the length of the bridge, if it is not uniform. This theoretical model successfully predicts the gross features of the I-V curve. It predicts that for large dc power (i.e. large bias voltages), the electrons are heated well above Tc , and the I-V curve approaches the normal state (linear) I-V curve at high bias. Experimentally, we nd that the curves never meet, which may be due to the fact the some fraction of the electrons near the ends of the device are below
Tc, so that the resistance is always less than the normal state resistance by some
fraction. The theory also predicts a supercurrent branch, where there is no dc power dissipated, hence no heating. (The model does not, however, predict the value of the critical current.) Finally, the model predicts that near the dropback region, the electrons are near resistance with temperature is largest at
Tc .
Since the change in
Tc ,
the device is expected to have the best conversion
eciency in this region. This is indeed found to be the case experimentally, as will be discussed further below. The model fails to give quantitative agreement with the experimental data for several reasons. In the case of devices longer than the electron-phonon interaction length
Le0ph ,
the temperature
pro le is expected to be uniform, and the relationship between the electron temperature, bath temperature, and input power in the normal state can be measured using noise-thermometry, as discussed in section 5.3. However, for the long device measured in this thesis, the I-V curve is not smooth at any temperature. An example of this can be seen in gure 5.37. There, \kinks" in the I-V curve are very suggestive of the formation of local \hot-spots" (Skocpol et al. 1974). It is postulated that some regions of the device are in the normal state, hence dc power is dissipated there, sustaining an elevated temperature. Other regions may be superconducting. Since no power is dissipated in
75
those regions, the electrons remain at the bath temperature and hence remain superconducting. It is also possible that the kinks are due to phase slip centers (Skocpol 1974), which normally occur in one dimensional superconducting wires. Since the width of the devices studied in this thesis is larger than the superconducting coherence length, they are classi ed as two dimensional superconductors, making the analysis of the current density and critical current much more complicated, especially under non-equilibrium conditions. For the devices shorter than
Le0ph , the temperature pro le is expected to be non-uniform.
The
ends are expected to be near the bath temperature, and the center is at or above
Tc if the device is
biased at non-zero voltage. Therefore, in order to accurately predict the I-V curve, the temperature pro le would need to be self-consistently calculated. Presumably, such a calculation would need to include the assumption that dc power is only dissipated in normal regions, and that regions below Tc would not be heated. An additional complication arises from the fact that the devices with length less than
Le0ph
are only 1-5 x
Lee
in length, so that a local temperature could only be de ned
over 1-5 \pixels." A quantitative prediction of I-V curves remains an unsolved problem for future research. The electrons can be heated above the bath temperature with ac power as well as dc power. In gure 5.2, the I-V curve at a xed bath temperature of 2 K is plotted for various LO powers. The LO frequency was 20 GHz, which is too fast for the electronic system to follow. Therefore, there is a net rise in the electron temperature, but no substantial oscillatory component. With suciently large LO power, all the electrons can be heated above
Tc
into the normal state, and the I-V curve
becomes linear. (This case is not illustrated in gure 5.2.) The critical current (i.e. the maximum current the device can sustain with no voltage drop) decreases with increasing in LO power. This trend is not continuous, and at a certain LO power, the critical current jumps discontinuously to zero, and the I-V curve is no longer hysteretic. We have termed the case where just enough LO power is applied to completely suppress the critical current the \overpumped" case. This is because the LO power required for optimum gain is 2-3 dB the critical current.
less
than that required to completely suppress
76
100
2K 2.5 K
3K
50
3.5 K 4K
Currernt (µA)
0
5K 5.5 & 6 K
4.5 K
-50
-100 -2 -1 0 Voltage (mV) 1 2
Figure 5.1: I-V curves at di erent bath temperatures for device A1, with no LO power applied.
77
40
No LO
30
Current (µA)
-15 dB -5 dB -3 dB -2 dB -1 dB
20
10
+2 dB
+1 dB
0 dB
Tags refer to amount of LO power applied. 0 dB reference is power required for optimum gain.
0 0.0 100 80
Current (µA)
0.2
0.4
0.6 0.8 1.0 Voltage (mV)
1.2
1.4
60 40 20 0 0.0
0.5
1.0
1.5 2.0 Voltage (mV)
2.5
3.0
Figure 5.2: I-V curves under di erent LO powers for device A1. The bath temperature was 2 K.
78
5.1.2
Gain and noise vs. voltage, LO power
The measurements of the gain and noise were all performed at a bath temperature of 2 K, for several reasons. First, it was found that the conversion eciency improved by about 1 dB for devices A1 and B upon reducing the temperature from 4.2 K to 2 K. Second, the conversion eciency and noise both had very non-uniform dependence on the dc bias voltage until the temperature was reduced to below approximately 3 K. The I-V curves were also smooth below that temperature, but developed kinks above that temperature, as can be seen in gure 5.1. (This was not the case for device E, which had kinks in the I-V curves at all temperatures measured, from 2 K to
Tc .)
This would have
complicated the analysis as well as the search for optimum bias points at a given temperature. It was also desired to compare di erent devices under nominally similar conditions, which was much simpler using the smooth I-V curves below 3 K. Finally, experiments at JPL at 500 GHz on very similar di usion-cooled devices found an improvement in the receiver noise temperature by about a factor of two for the same reduction in bath temperature from 4.2 K to 2 K (Skalare et al. 1996). This is in accord with the theoretical prediction of equation 2.31, since more LO power is required to heat electrons to Tc from a lower bath temperature, and the thermal conductance is reduced linearly with the bath temperature for di usion cooling. (The electron temperature is presumed to be near
Tc for mixing to occur.)
Therefore, unless otherwise speci ed, the measurements in this thesis were
undertaken with a bath temperature of 2 K. The (relative) conversion eciency, output noise, and mixer noise are plotted as a function of LO power for xed dc voltage in gure 5.3 for device A1. There are two cases of LO power which are of interest. We refer to the LO power required to maximize the (coupled) conversion eciency as the \optimum gain" case. Note that the conversion eciency and output noise peak at di erent LO powers, for a xed bias voltage. However, the
mixer
noise is relatively constant near its minimum,
even though the eciency and output noise are changing very rapidly with LO power there. The second qualitative case is the \overpumped" case, where the critical current is suppressed. In
that case, the output noise is drastically suppressed relative to its maximum value. The conversion eciency is also somewhat lower than its maximum value. However, the mixer noise does not change much between the \optimum gain" case and the \overpumped" case. It is possible to explain this as consistent with equations 2.13 and 2.28. The eciency is proportional to the LO power, while the
79
output noise does not depend on it directly. Therefore, it is conceivable that the eciency peaks at a higher LO power than the output noise. The overpumped case is of practical interest because the output noise and eciency are less sensitive to the dc bias voltage, which will be discussed next. The general behavior indicated in gure 5.3 was observed in all the devices measured. With the exception of device E, the
mixer
noise in the overpumped case at the dc bias that minimized the
mixer noise was lower than the mixer noise in the optimum gain case at the dc bias that minimized the mixer noise. An important practical issue is the input power which saturates the detector. In gure 5.4, the relative gain is plotted as a function of input power for device B. The 1 dB gain compression point is approximately 1/3 of the LO power, and the curve indicates that the device saturates very smoothly with increasing signal power. Although the saturation curve was not measured for each of the devices studied in this thesis, it is reasonable to postulate a scaling law for the saturation power. The LO power (in addition to the dc power) provides an indication of how much power must be applied to bring the electron temperature to
Tc .
It is then plausible that the saturation signal
power is approximately 1/3 of the LO power for all devices. In order to investigate the dependence of the gain and noise on dc bias, the output noise and conversion eciency were measured as a function of dc bias for two di erent LO powers (optimum gain, and overpumped) for each device. The I-V curve was also simultaneously measured. The resultant mixer noise was calculated by taking the ratio of the output noise to the conversion eciency. The measurements were done at an intermediate frequency that is low enough to be representative of the zero IF limit of the device performance. The results are plotted in gures 5.55.24. The di erential and absolute resistance are also plotted with the corresponding I-V curves, since they will be needed for the data analysis. (Note that both quantities are between 20
and 130
for all of the devices, except E. This means they were all well coupled to the rf and IF system. This point will be discussed in more detail in section 5.2.) The immediate conclusion in these graphs is that the mixer noise is
very low,
varying from
100 to 500 K (DSB) for all of the data. Since 1 THz.
the mixing process is thermal and depends only on heating of the electrons, similar results are expected to be achievable at THz frequencies. This indicates that the achievable receiver or system noise is at least an order of magnitude below existing technologies for frequencies above
80
Excellent results at THz frequencies have already been achieved in experiments on di usion-cooled hot-electron bolometers, and these will be discussed further in chapter 6. In addition to the low mixer noise, we have observed two general trends. First, in the optimum gain case, the conversion eciency and output noise rise sharply near the point of instability where the device can switch to the supercurrent branch. The one can bias near the region of instability.
mixer
noise gets slowly lower, the closer
Second, in the overpumped case, the output noise
mixer
and conversion eciency are reduced compared to the optimum gain case, and the
noise is
usually lower. Additionally, since the I-V curve is non-hysteretic, the device cannot switch into the superconducting branch and the eciency and noise depend smoothly on the dc bias. (Note that in the overpumped case the device sometimes switches into the superconducting state because not quite enough LO power was applied.) In section 5.4, these data will be compared with theoretical predictions based on chapter 2. The measured output noise (from 175-215 MHz) for device A1 in gures 5.5 - 5.7 was done with an LO frequency of 18 GHz. When the output noise was measured for the same device with an LO frequency of 20 GHz over a broader band, the output noise was about a factor of 2 higher in the optimum gain case, but a factor of 2 lower in the overpumped case. Additionally, the conversion eciency plotted in gures 5.5 - 5.7 was measured with an LO frequency of 18 GHz as a
relative
quantity. It has been normalized to agree with the conversion eciency measured over a broader IF band with an LO of 20 GHz. (The conversion eciency measured with a 20 GHz LO was also calibrated in a slightly di erent manner than the other devices: incoherent calibrations were used to determine the coupled rf power as well as the gain of the IF ampli ers.) Therefore, the mixer noise and conversion eciency for device A1 in gures 5.5 and 5.7 are valid as relative quantities, but have been normalized in an indirect manner, and are therefore less trustworthy. Additionally, the periodic variation of the output noise and eciency with bias voltage is not an experimental artifact. The period is approximately 35
V , which corresponds to a frequency of 8.5 GHz (using = eV ). h!
However, the spacing of the peaks did not scale with the applied LO frequency. The periodicity in the
mixer noise
may be an experimental artifact, since the o set voltage in the dc electronics
may have shifted by an amount comparable to the periodicity between the measurements of the conversion eciency and the measurements of the output noise. This behavior was only observed
81
for device A11 , and so may be a mesoscopic e ect related to the fact that the length of the device was comparable to
Lee .
1
The bias voltage dependence of the gain and noise for device A2 was not carefully measured.
82
60
0
Output noise Efficiency
Output noise (K), Mixer noise
50 40
-4
Eff. (dB, rel. units)
-8 30
"Overpumped"
-12 -16 -20 15
20 10 0 -30 -25
Mixer noise Arb. units
-20
-15 -10 -5 0 LO power (dB, rel. units)
5
10
Figure 5.3: Eciency, output and mixer noise for device A1 vs. LO power. The optimum gain occurs at PLO = 0 dB. Bias voltage = 0.6 mV. Noise and eciency measured at IF = 1.25-1.75 GHz.
5
LO Power = 10 nW
Relative Gain (dB)
0
-5
1 dB Gain Compression Point @ 3.3 nW
-10
-15 0.01
0.1
1 Input RF Power (nW)
10
100
Figure 5.4: Saturation curve for device B.
83
4
Conversion efficiency (SSB)
2
Efficiency IF = 125 - 215 MHz
4 2
Noise temperature (K)
0.18
6 4 2
Output noise IF = 125 - 215 MHz Mixer noise (DSB) IF = 125 - 215 MHz
8 6 4 2 8 6 4 2
1000
0.018
6 4 2
100
0.001
-1.2
-0.8
-0.4 0.0 0.4 Voltage (mV)
0.8
1.2
10
Figure 5.5: Gain and noise vs. voltage for device A1 in overpumped case.
100 40 80 20
Current
Resistance (Ω)
Current (µA)
60
dV/dI
0 -20
40 20 0
Vdc/Idc
-40 -1.2 -0.8 -0.4 0.0 0.4 Voltage (mV) 0.8 1.2
Figure 5.6: Current vs. voltage for device A1 in overpumped case.
84
4
4
Efficiency IF = 125 - 215 MHz Mixer noise (DSB) IF = 125 - 215 MHz
Conversion efficiency (SSB)
2
2
Noise temperature (K)
0.18
6 4 2
8 6 4 2 8 6 4
1000
0.018
6 4 2
Output noise IF = 125 - 215 MHz
100
2
0.001
-1.2
-0.8
-0.4 0.0 0.4 Voltage (mV)
0.8
1.2
10
Figure 5.7: Gain and noise vs. voltage for device A1 in optimum gain case.
100 40 20 0 40 -20
Vdc/Idc Current
80
Resistance (Ω)
Current (µA)
dV/dI
60
20 0
-40 -1.2 -0.8 -0.4 0.0 0.4 Voltage (mV) 0.8 1.2
Figure 5.8: Current vs. voltage for device A1 in optimum gain case.
85
4
4
Bias point for used for measurements
Conversion efficiency (SSB)
2
2
Noise temperature (K)
0.18
6 4 2 Efficiency
IF = 125-215 MHz
Mixer noise (DSB) IF = 125-215 MHz
8 6 4 2 8 6 4 2
1000
0.018
6 4 2
100
Output noise IF = 125-215 MHz
0.001 -2.0
-1.5
-1.0
-0.5 0.0 0.5 Voltage (mV)
1.0
1.5
10 2.0
Figure 5.9: Gain and noise vs. voltage for device B in overpumped case.
100 40 80 20
dV/dI
Resistance (Ω)
Current (µA)
60 40
0 -20 -40 -2 -1
Current Vdc/Idc
20 0
0 Voltage (mV)
1
2
Figure 5.10: Current vs. voltage for device B in overpumped case.
86
4
4
Bias point used for measurements
Conversion efficiency (SSB)
2
2
Noise tempeature (K)
0.18
6 4 2
Efficiency IF = 125-215 MHz
Mixer noise (DSB) IF = 125-215 MHz
8 6 4 2 8 6 4 2
1000
0.018
6 4 2
100
Output noise IF = 125-215 MHz
0.001 -2.0
-1.5
-1.0
-0.5 0.0 0.5 Voltage (mV)
1.0
1.5
10 2.0
Figure 5.11: Gain and noise vs. voltage for device B in optimum gain case.
140 60
dV/dI
120 100
40
Resistance (Ω)
Current (µA)
20 80 0 60 -20 40 -40
Current Vdc/Idc
20 -1.0 -0.5 0.0 0.5 Voltage (mV) 1.0 1.5 2.0 0
-60 -2.0 -1.5
Figure 5.12: Current vs. voltage for device B in optimum gain case.
87
4
4 2
Conversion efficiency (SSB)
2
Noise temperature (K)
0.18
6 4 2 Efficiency
IF = 1.25-1.75 GHz Mixer noise (DSB) IF = 1.25-1.75 GHz
8 6 4 2 8 6 4 2
1000
0.018
6 4 2
Output noise IF = 1.25-1.75 GHz
100
0.001 -2.0
-1.5
-1.0
-0.5 0.0 0.5 Voltage (mV)
1.0
1.5
10 2.0
Figure 5.13: Gain and noise vs. voltage for device C in overpumped case.
140 40 20 0 -20
Current dV/dI
120 100 80 60
Vdc/Idc
Resistance (Ω)
Current (µA)
40 20
-40 -2.0 -1.5 -1.0 -0.5 0.0 0.5 Voltage (mV) 1.0 1.5 2.0
0
Figure 5.14: Current vs. voltage for device C in overpumped case.
88
4
4 2
Conversion efficiency (SSB)
2
Noise temperature (K)
0.18
6 4 2
Mixer noise (DSB) IF = 1.25-1.75 GHz
8 6 4 2
1000
0.018
6 4 2
Output noise IF = 1.25-1.75 GHz
Efficiency IF = 1.25-1.75 GHz
8 6 4 2
100
0.001 -2.0
-1.5
-1.0
-0.5 0.0 0.5 Voltage (mV)
1.0
1.5
10 2.0
Figure 5.15: Gain and noise vs. voltage for device C in optimum gain case.
200 80 160 40 120 0 -40
Vdc/Idc Current dV/dI
Resistance (Ω)
Current (µA)
80 40 0
-80 -2.0 -1.5 -1.0 -0.5 0.0 0.5 Voltage (mV) 1.0 1.5 2.0
Figure 5.16: Current vs. voltage for device C in optimum gain case.
89
4
Conversion efficiency (SSB)
2
Bias point used for measurements Efficiency IF=125-215 MHz
Noise temperature (K)
0.1
6 4 2
1000
0.01
6 4 2
Mixer noise (DSB) IF=125-215 MHz
100
0.001
6
Output noise IF=125-215 MHz
10 1.5 2.0
-2.0
-1.5
-1.0
-0.5 0.0 0.5 Voltage (mV)
1.0
Figure 5.17: Gain and noise vs. voltage for device D in overpumped case.
100 40 20 60 0
Current Vdc/Idc dV/dI
80
Resistance (Ω)
Current (µA)
40 20
-20 -40 -2.0 -1.5 -1.0 -0.5 0.0 0.5 Voltage (mV) 1.0 1.5 2.0 0
Figure 5.18: Current vs. voltage for device D in overpumped case.
90
4
4
Efficiency IF=125-215 MHz
Conversion efficiency (SSB)
2
2
Noise temperature (K)
0.18
6 4 2
Bias point used for measurements
8 6 4 2 8 6 4 2
1000
0.018
6 4 2
Output noise IF=125-215 MHz
100
Mixer noise (DSB) IF=125-215 MHz
0.001 -2.0
-1.5
-1.0
-0.5 0.0 0.5 Voltage (mV)
1.0
1.5
10 2.0
Figure 5.19: Gain and noise vs. voltage for device D in optimum gain case.
300 200
dV/dI
140 120 100
Current
Resistance (Ω)
Current (µA)
100 0
80 60
-100 -200 -300 -2.0 -1.5 -1.0 -0.5 0.0 0.5 Voltage (mV) 1.0 1.5
Vdc/Idc
40 20
2.0
0
Figure 5.20: Current vs. voltage for device D in optimum gain case.
91
4
4 2
Conversion efficiency (SSB)
2
Noise temperature (K)
0.18
6 4 2
Mixer noise (DSB) IF=125-215 MHz Efficiency IF=125-215 MHz
8 6 4 2 8 6 4
1000
0.018
6 4 2
Output noise IF=125-215 MHz
100
2
0.001
-8
-6
-4
-2 0 2 Voltage (mV)
4
6
8
10
Figure 5.21: Gain and noise vs. voltage for device E in overpumped case.
200 150 100
Bias point used for measurements Current
300 250 200
Resistance (Ω)
Current (µA)
50 0 -50 -100 -150 -200 -8 -6 -4
dV/dI
150 100 50
Vdc/Idc
0 -50
-2 0 2 Voltage (mV)
4
6
8
-100
Figure 5.22: Current vs. voltage for device E in overpumped case.
92
4
4
Efficiency IF=125-215 MHz
Conversion efficiency (SSB)
2
2
Noise temperature (K)
0.18
6 4 2
8 Mixer noise (DSB) 6 IF=125-215 MHz 4
Output noise IF=125-215 MHz
1000
2 8 6 4 2
0.018
6 4 2
100
0.001
-8
-6
-4
-2 0 2 Voltage (mV)
4
6
8
10
Figure 5.23: Gain and noise vs. voltage for device E in optimum gain case.
200 150 100
dV/dI Current
300 250 200
Resistance (Ω)
Current (µA)
50 0 -50 -100 -150 -200 -8 -6 -4 -2 0 2 Voltage (mV) 4 6 8
Vdc/Idc Bias point used for measurements
150 100 50 0 -50 -100
Figure 5.24: Current vs. voltage for device E in optimum gain case.
93
5.1.3
Gain vs. intermediate frequency
One of the most important goals of this work was to investigate the dependence of the conversion eciency on the intermediate frequency and determine the time constant as a function of device length. The dependence of the conversion eciency on intermediate frequency is plotted for all the devices measured in gure 5.25. The absolute conversion eciency will be discussed further in section 5.4. However, in order to compare the relative frequency response of each device, the relative frequency dependence is plotted in gure 5.25. The theoretical ts to equation 2.13 are also shown. The frequency dependence of the conversion eciency is indeed well-described by equation 2.13. Note that there are two devices of the shortest length plotted, and the data are very consistent. The close agreement between the theory and experiment provides strong con rmation of the theoretical model over two orders of magnitude in frequency and conversion eciency. The tted time-constant is plotted in gure 5.26 as a function of length. This plot is the central result of this thesis. When the device length
L
is much larger than
Le0ph
(
1 m at 4.2 K),
the bandwidth is expected to be independent of length. The dashed line indicates this phonon cooling limit. Device E is in this limit. For be di usion, and the dotted line shows
L Le0ph , the dominant cooling mechanism should the expected L02 dependence. The solid line shows the
prediction for the net e ect of both phonon and di usion cooling mechanisms, assuming the thermal conductances add. The theoretical prediction for the di usion cooling based on equation 2.57 is that th (ns)
L , with L in m. We nd experimentally that
2
th (
ns) 1:8 L2 .
This discrepancy
appears to be within the uncertainties in the predicted as well as the measured prefactor. While the prediction for th is exact for a non-superconducting bridge, it also can be used below the quasiparticle excitations which carry away heat have energy gap energy near
kT which is much larger than the
Tc
because
Tc .
The measured bandwidth of 6 GHz is the largest bandwidth yet obtained in a
low-Tc bolometric mixer, as of the writing of this thesis. The value of 6 GHz is actually a lower limit, since since the conversion eciency changes with IF by an amount comparable to the experimental uncertainties for the IF frequencies used. It is possible that the measured time constant is modi ed by electro-thermal feedback e ects, and that the \bare" time constant is di erent from the measured one. However, in section 5.4, the \slowing factor" ( ) is estimated, and for all the devices it is less than 0.25, with the exception of
94
device E. For device E,
is 0.46 in the optimum gain case.
Therefore, the inferred time constant is
approximately equal to th .
5.1.4
Noise vs. intermediate frequency
In the previous subsection, it was demonstrated that di usion cooling could be used to increase the conversion eciency bandwidth by a factor of 50. An additional quantity of practical interest is the mixer noise, i.e. the noise referred to the input. If the output noise decreases with intermediate frequency in the same way as the conversion eciency, then the mixer noise will be independent of the intermediate frequency. For this reason, the spectrum of the output noise was measured under identical conditions as the conversion eciency. Measurements at microwave frequencies of noise are usually performed with an isolator between the device under test and the ampli er. The isolator serves to insure that noise generated by
the ampli er does not get re
ected o of the device under test back into the ampli er input, and mistaken as noise from the device. Broadband, coolable isolators are not available, so most of the measurements presented in this section were done without an isolator. The re
ection coecient of each device was measured under the same conditions that the noise was measured under, and the devices were found to be well coupled in all cases, except for device E at frequencies below 100 MHz. (The re
ection measurements are discussed in more detail below in section 5.2.) Therefore, the lack of an isolator should not introduce signi cant errors into the measurement process. However, in several cases an isolator was used in order to con rm that the measured noise with and without an isolator agreed. This is addressed in gure 5.27, where the noise in the band 5-7.5 GHz is plotted vs. dc bias, measured in two separate experiments, one with an isolator and one without. The results agree, and it was found that the drift in the calibration coecients was a larger source of uncertainty than that introduced by not using an isolator. The output noise for each device was measured as a function of frequency in the case of optimum gain and in the overpumped case. In both cases, the dc bias was chosen to minimize the mixer noise. The calibration technique was described in section 3.5. The output noise was measured under identical conditions (usually during the same experiment) as the conversion eciency. Then, the mixer noise was calculated by dividing the measured output noise at each frequency by the conversion
L = 0.08 µm (two devices)
0
-5
L = 0.16 µm
-10
-15
L = 3 µm L = 0.6 µm L = 0.24 µm
Relative Conversion Efficiency (dB)
-20
Experimental data Theoretical fits
2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9
Figure 5.25: Relative conversion eciency vs. intermediate frequency for devices of di erent length.
-25 4 0.1
5 6 7 8 9
1 Intermediate Frequency (GHz)
10
95
96
2
8
10 10
8 7 6 5 4 3
Measured time constant Diffusion cooled scaling Phonon cooled scaling Diffusion plus phonon cooling Time constant measured by Skalare using 500 GHz rf signal
2 3 4 5 6 7 8
IF Bandwidth (GHz)
Time constant (ps)
2
100 1
8 7 6 5 4 3 2 2 3 4 5 6 7 8
1000 0.1
8 7 6 5 4 2 3 98 7 6 5
3
2
1
98 7 6 5
4
3
2
0.1
Bridge length (µm)
Figure 5.26: Bandwidth vs. device length.
97
eciency at that frequency. The results of these measurements are plotted in gures 5.28-5.35. The points for frequencies above 1 GHz are averaged over a 500 MHz bin, and the points for frequencies below 1 GHz are averaged over a 100 MHz bin. Additionally, theoretical ts to the functions
Tout(!IF ) = TJohn: +
and
TT:F: (0) 1 + (!IF ef f NOISE )2
(5.1)
2 Tmix(!IF ) = a + b!IF
(5.2)
are plotted. Note that the 3 dB gain bandwidth (i.e. the frequency at which the gain falls by a factor of 2) is predicted to be (2 ef f )01 , and the frequency at which the thermal
uctuation noise component of the output noise falls by a factor of two is predicted to be the
same,
i.e. (2 ef f )01 . In order
to test this prediction experimentally, both quantities were varied in the ts to the measured conversion eciency and output noise. The results of these ts are summarized in tables 5.1 and 5.2. The relative spectrum of the output noise behaves similarly with frequency as the conversion eciency, as can be seen by comparing the tted time constant for the conversion eciency and output noise. This implies that the 3 dB noise bandwidth is larger than the 3 dB gain bandwidth, which is also indicated by comparing the two quantities in tables 5.1 and 5.2. The noise bandwidth is larger than the gain bandwidth by a factor of 1.4-9.4. The tted white component of the output noise should be approximately the electron temperature, which is presumed to be near
Tc
5:5
K.
However,
the tted value is larger than this, from 8-25 K. For the di usion cooled devices, it is dicult to determine the white portion accurately, since the frequency-dependent thermal
uctuation noise is a signi cant component of the total output noise at all frequencies within the measurement band. However, for device E it was possible to measure the output noise at 4.75-7.25 GHz, which is higher than the thermal time constant of
100 MHz for that device. There, the dominant noise should
no longer be the thermal
uctuation noise. In gure 5.36, the output noise (measured with an isolator) from 4.75-7.25 GHz is plotted vs. bias voltage for a bath temperature of 2 K, and a bath temperature of 7 K. For the 7 K data, the device is in the normal state, and the increase in noise with increasing bias voltage is due to the fact that the electron temperature (and hence the Johnson noise) rises with increasing dissipated
98
dc power. The 2 K data shows very similar behavior with bias voltage, except near the dropback. Near the dropback, the thermal
uctuation noise is evident, but only at the level of about 0.5 K. The other component of the noise is interpreted as Johnson noise, due to the fact that the electrons are hotter than the bath. Therefore, it appears that the high-frequency (i.e. is indeed only due to Johnson noise, and not some other noise source. In gure 5.34, where the output noise for device E is plotted against the intermediate frequency, it is clear that the output noise is still rising below 100 MHz, the lowest frequency measured in that experiment. Therefore, in a separate experiment the noise of device E was measured from 1825 MHz. (The noise temperature of the rst ampli er was 300 K at that frequency.) In gure 5.37, the output noise at 20 MHz is plotted vs. bias voltage. In contrast to the other devices, the output noise depends very sensitively on the bias voltage. The peaks in the output noise correspond in many cases to regions of negative di erential resistance on the I-V curve, which is also plotted in gure 5.37. The spectrum of the output noise at bias points near regions of negative di erential resistance was not white, indicating that the device may be oscillating in the MHz frequency region. The measured (ampli ed) noise power, including the ampli er input noise, is plotted vs. frequency in gure 5.38. There, it is clear that the output noise is not white; this behavior was not observed for any of the other devices. Thus, since device E has regions of negative di erential resistance on the I-V curve, it tends to oscillate, which can dominate the thermal
uctuation noise at frequencies
f
0 ) component
th
1
0 below th1 for that device. The physical mechanism for the negative di erential resistance is an open
subject for future research.
99
50 40
50 40
< Tout > 5-7.5 GHz (K)
Current (µA)
Current
30 20 10
Noise measured without isolator Noise measured with isolator
30 20 10 0
0 0.0
0.1
0.2
0.3
0.4 0.5 0.6 Voltage (mV)
0.7
0.8
0.9
1.0
Figure 5.27: Output noise vs. bias voltage for device A1, with and without isolator, in optimum gain case. The error bars are the variances in the output noise across the band.
100
2
Optimum gain Data Theory, fit
100
Output noise (K)
10
6 5 4 3 2
6 5
Overpumped
0.1
2
3
4
5 6 7 8 9
1 Frequency (GHz)
2
3
4
5 6 7 8 9
10
Figure 5.28: Output noise vs. frequency for device A1. In the overpumped case, the output noise was measured at the bias voltage which maximized Tout ; the conversion eciency was not measured in the overpumped case.
1000
8 7 6 5 4 3
Optimum gain
Mixer noise (K, DSB)
Data Theory, fit
2
100
8 7 6 5
0.1
2
3
4
5 6 7 8 9
1 Frequency (GHz)
2
3
4
5 6 7 8 9
10
Figure 5.29: Mixer noise vs. frequency for device A1.
101
100
8 7 6 5
Optimum gain
Output noise (K)
4 3 2
Overpumped Point measured using isolator
10
8 7 6 5
Data Theory, fit
0.1
2
3
4
5 6 7 8 9
1 Frequency (GHz)
2
3
4
5 6 7 8 9
10
Figure 5.30: Output noise vs. frequency for device B.
2
Mixer noise (K,DSB)
1000
8 7 6 5 4 3 2
Overpumped Data Theory, fit Optimum gain
100 0.1
2
3
4
5 6 7 8 9
1 Frequency (GHz)
2
3
4
5 6 7 8 9
10
Figure 5.31: Mixer noise vs. frequency for device B.
102
2
100
Output noise (K)
Optimum gain
6 5 4 3 2
Optimum gain, measured using isolator
10
6 5
Data Theory, fit
Overpumped
Overpumped, measured using isolator
0.1
2
3
4
5 6 7 8 9
1 Frequency (GHz)
2
3
4
5 6 7 8 9
10
Figure 5.32: Output noise vs. frequency for device D.
105
6 4
Mixer noise (K, DSB)
2
104
6 4 2
Data Theory, fit
103
6 4 2
Overpumped
Optimum gain
102 0.1
2
3
4
5 6 7 8 9
1 Frequency (GHz)
2
3
4
5 6 7 8 9
10
Figure 5.33: Mixer noise vs. frequency for device D.
103
2
100
Output noise (K)
7 6 5 4 3 2
Overpumped
Data Theory, fit Optimum gain
Optimum gain, measured with isolator
10
7 6 5 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9
0.1
1 Frequency (GHz)
10
Figure 5.34: Output noise vs. frequency for device E.
Mixer noise (K,DSB)
105
Overpumped
104
103
Optimum gain Data Theory, fit
102 0.1
2
3
4
5 6 7 8 9
1 Frequency (GHz)
2
3
4
5 6 7 8 9
10
Figure 5.35: Mixer noise vs. frequency for device E.
104
Device
L (m)
Gain BW (GHz)
Tout (0)
(K) 49 57 55 281 231
(0)
(dB) -5.6 -11 -8 -4.1 -2y
Tmix (0)
(K,DSB) 120 320 200 120 530
1 2 effNOISE
Noise BW (GHz) 30 3.9 0.73 0.75
TJohn:
(K) 25 23 19 8
(GHz) 2.3 1.4 0.13 0.13
A1 A2 B C3 D E
0.08 0.08 0.16 0.24 0.6 3
>6 >6
2.4 1.5 0.3 0.08
3 1.25-1.75 GHz. The spectrum was not measured for this device. y The lowest eciency measured was only -4 dB, but the t returned a value of -2 dB because the
lowest IF measured for this particular experiment was only 100 MHz.
Table 5.1: Fitted gain and noise bandwidths, optimum gain case.
Device
L (m)
Gain BW (GHz)
Tout (0)
(K) 13 16 16 49 69
(0)
(dB) -7 -13.5 -12.7 -10.4 -11.7
Tmix (0)
(K,DSB) 170 160 120 310
1 2 effNOISE
Noise BW (GHz) 3.1 0.53 0.16
TJohn:
(K) 10 16 7
(GHz) 2.3 0.11 0.045
A1 A2 B C3 D E
0.08 0.08 0.16 0.24 0.6 3
>6
2.25 1.5 0.38 0.064
3
Table 5.2: Fitted gain and noise bandwidths, overpumped case. 1.25-1.75 GHz. The spectrum was not measured for this device.
105
12
Current
400
< Tout > 4.75-7.25 GHz (K)
11 200 10 9 8 7
Output noise; bath temperature 2 K Output noise; bath temperature 7 K, (device in normal state at all bias voltages)
Current (µA)
0
-200
6 -30
-20
-10
0 10 Voltage (mV)
20
-400 30
Figure 5.36: Output noise vs. voltage at 6 GHz for device E. (The slight skew of the 2 K data is probably due to drift of the calibrations during the measurement.) The measurements were done with no LO applied, but the application of LO power did not change the results taken at a bath temperature of 2 K by more than 1 K at any bias voltage.
106
1000
8 6 4
Output noise
160 140
< Tout > 18-25 MHz (K)
Current (µA)
2
120 100
Current
100
8 6 4 2
80 60 10
10
0
1
2
3
4 5 6 Voltage (mV)
7
8
9
Figure 5.37: Output noise vs. voltage at 20 MHz for device E.
0 -10
Power (dB, arb. units)
-20 -30 -40 -50 -60 0 10 20 Frequency (MHz) 30 40
Figure 5.38: Output noise vs. frequency below 40 MHz for device E.
107
5.1.5
Noise vs. LO frequency
Since the LO frequency was under experimental control, the output noise was measured when the applied LO signal frequency was both above and below (2 th )01 . If the LO signal frequency is higher than (2 th )01 , the electron temperature will not vary with time. Rather, there will be a net increase in the temperature. On the other hand, if the LO signal frequency is lower than (2 th )01 , then the electron temperature may vary with time. The theoretical description of the thermal
uctuation noise in this case has not been developed, but was measured experimentally nonetheless. The measured output noise at 1.25-1.75 GHz vs. bias voltage is plotted in gure 5.39 in the overpumped case for LO frequencies between 4 and 40 GHz for device A1. (At each LO frequency, just enough LO power was applied to completely suppress the critical current.) It is clear from this gure that the LO power causes an enhancement of the noise for frequencies below suppression of the noise at frequencies above
10 GHz. In order to indicate the dependence of the
10 GHz, and a
output noise on the applied signal frequency, we have plotted the output noise (at the dc bias which maximized the output noise) as a function of the applied LO frequency in gure 5.40 for devices A1 and B. For device B, there is no increase in the output noise for LO frequencies above 8 GHz. Note that 8 GHz is already 3 times larger than (2 th )01 for device B. Therefore, for both devices, the output noise is suppressed when a large signal with frequency higher than (2 th )01 is applied. Additionally, the noise is dramatically enhanced for device A1 when a large signal with frequency lower than (2 th )01 is applied.
108
100
8 GHz Current, 20 GHz 6 GHz 4 GHz
104
Output noise (K), 1.25-1.75 GHz
50
103
Current (µA)
0
40 GHz
102
-50
Current, 8 GHz 20 and 30 GHz Textboxes refer to frequency of applied signal
101
-100 -3
-2
-1
0 1 Voltage (mV)
2
3
100
Figure 5.39: Output noise vs. bias voltage for device A1 in the overpumped case, for di erent LO frequencies.
2
Output noise (K), 1.25-1.75 GHz
1000
8 6 4
Device A1
2
100
8 6 4 2
Device B
10
0
10
20 30 Applied RF Frequency (GHz)
40
Figure 5.40: Output noise vs. LO frequency for devices A1 and B in overpumped case.
109
5.2
Device impedance measurements
In this section, the impedance measurements of the device in the mixed state are described. The results of these measurements are important for several reasons. First, the coupling between the device and the ampli er depends on the device impedance. The calibration technique used for the noise measurements are especially sensitive to the device impedance. Second, it is important to test the theoretical prediction for the device impedance discussed in section 5.2, which is the most fundamental aspect of the theory of hot-electron bolometers. Third, impedance measurements of the device in the normal state provide information about the electromagnetic coupling to the device from the 50
system. This should be resonance free if frequency-dependent device parameters are to be studied. The quantity actually measured in this thesis is the return loss o of the device, i.e. the (power) re
ection coecient of an ac signal propagating down toward the device. The (voltage) re
ection coecient 0 is related to the load impedance of the device terminating a transmission line by the well known formula: 0= where
ZL 0 Z0 ; ZL + Z0 Z0
(5.3)
ZL
is the load (i.e. device) impedance, and
is the characteristic impedance of the trans-
mission line, which was 50
for the systems used in this thesis work. The return loss, i.e. the ratio of the power incident on the device to the power re
ected o of the device, is usually expressed in decibels as:
RL = 020 logj0j (dB ):
(5.4)
In this measurement, a directional coupler (see gure 3.1) allows a weak signal to be propagated down toward the device. When the device is biased in the superconducting state, all of the power is re
ected o of the device, coupled into the ampli er, ampli ed, and then measured on the spectrum or network analyzer. This serves as a reference calibration. The ratio of the ampli ed and re
ected power so measured on the spectrum analyzer when the device is in the intermediate state to the power so measured when the device is in the superconducting state provides a measure of the power re
ection coecient. This technique neglects the imperfections of the coax connectors and the directional coupler, and so is only approximate. An exact calibration would require a more
110
Measured power (dB, arb. units)
-20
Dev. in superconducting state Dev. in normal state
-30
-40
Dev. in intermediate state
-50 0 1 2 3 4 Frequency (GHz) 5
6
7
Figure 5.41: Measured re
ected power for return loss. sophisticated treatment of the non-idealities of the system2 . Additionally, this method was only a
scalar
method, since only the magnitude of the re
ection coecient was measured, and not the
phase. In order to illustrate the technique, the raw measured powers are plotted in gure 5.41, measured using device D. These are the uncorrected data, and the ratio is calculated using these data to determine the return loss. The strong frequency dependence of the data below 1 GHz is a convolution of the strong frequency dependence of the coupling through the directional coupler and the gain of the ampli er chain. This makes the measurements below 1 GHz less accurate than the measurements above 1 GHz. The results of the return loss measurements are plotted in gures 5.42 - 5.47. The results
are plotted both for the device in the normal state, as well as in the intermediate state in the overpumped and optimum gain cases. From these plots, two important conclusions can be drawn.
2 A vector open, short, load calibration treatment was attempted, where the open, short, and load were placed at the end of the 1" section of stainless steel coax, cooled, and measured. The calibration was not reliable since the microwave properties of the system changed on thermal cycling. A more concerted e ort would have been needed to solve these and other associated calibration problems.
111
First, the devices are all well coupled in a broadband, resonance free manner to the 50
system in the normal state. (A 10 dB return loss corresponds to a 90% power coupling of an incoming signal to the device.) This means that the mounting technique used provides a good 50
transmission line system up to the device, without any unwanted parasitic capacitance or inductance. This is important because the main purpose of this thesis is to investigate the frequency dependence of the gain and noise in order to extract information about the underlying physical processes occurring in the device. The second conclusion is that, with the possible exception of device E, all of the devices were well-coupled to the 50
system when in the intermediate state. (Although the plots are only for one particular dc bias voltage, it was found that the dependence of the return loss on the bias voltage was very weak.) Additionally, these measurements are in accord with the theoretical prediction3 which states that the device impedance at
f < (2 th )01
should be
dV=dI , and at f > (2 th )01 Vdc =Idc.
The values of these two parameters and the predicted and measured return loss are summarized in tables 5.3 and 5.4. The values of Vdc =Idc and
dV=dI
fall within the range of 23-100
, which is close
enough to 50
according to equation 5.4 to allow power coupling of approximately 90%. Device E had values of
Vdc =Idc
between 10 and 20
, so that the power coupling to the 50
system was
only between 50 and 80%, theoretically as well as experimentally. The frequency dependence of this coupling was still smooth enough for the studies carried out in this thesis. Finally, the coupling to the device at frequencies above
12 GHz was more dicult to measure
because the gain of the ampli er used dropped rapidly above that frequency. However, a separate measurement of the return loss o of a 50
chip resistor in a similar mounting con guration at room temperature showed a resonance free coupling better than 90% from 50 MHz to 40 GHz. Although that con guration was slightly di erent, it seems reasonable to assume that the devices were well coupled to the 50
system up to 40 GHz.
3 In this section, we neglect the di erence between and =(1 0 ), which is the correct quantity to use for 0 th th the frequency of interest regarding the device impedance.
112
5 0
Dev. in normal state Dev. in intermediate state Optimum gain Overpumped
- Return loss (dB)
-5 -10 -15 -20 -25 0 1 2
3 4 5 Frequency (GHz)
6
7
8
Figure 5.42: Measured return loss for device A1.
5 0
- Return loss (dB)
-5 -10 -15 -20 -25 0 1 2 3
Dev. in normal state
4 5 6 7 Frequency (GHz)
8
9
10
11
Figure 5.43: Measured return loss for device A2.
113
5 0
Dev. in intermediate state Optimum gain Overpumped
- Return loss (dB)
-5 -10 -15 -20 -25
Dev. in normal state
0
1
2
3 4 5 Frequency (GHz)
6
7
8
Figure 5.44: Measured return loss for device B.
5 0
Dev. in intermediate state Optimum gain
- Return loss (dB)
-5 -10 -15 -20 -25 0 1 2 3 4 5 6 7 8 Frequency (GHz) 9 10 11 12
Figure 5.45: Measured return loss for device C.
114
5 0
Dev. in normal state
- Return loss (dB)
-5 -10 -15 -20 -25 0 1 2 3 4 Frequency (GHz)
Dev. in intermediate state Optimum gain Overpumped
5
6
7
Figure 5.46: Measured return loss for device D.
5 0
Dev. in intermediate state Optimum gain Overpumped
- Return loss (dB)
-5 -10 -15
Msmt. noise floor
-20 -25 0.0 0.5
Dev. in normal state
1.0
1.5 2.0 Frequency (GHz)
2.5
3.0
Figure 5.47: Measured return loss for device E.
115
Device
L (m)
RN
(
)
dV/dI (
)
RL theory (dB)
RL expt. (dB)
Vdc =Idc
(
)
RL theory (dB)
RL expt. (dB)
0 ! < th1
A1 B C D E 0.08 0.16 0.24 0.6 3 56 80 96 93 86 32 90 100 83 65 13 11 10 12 18
0 ! < th1
0 ! > th1
23 28 40 28 11 9 11 19 11 4
0 ! > th1
>10 >10 >10
8 -
>10 >10 >10 >10
4
Table 5.3: Device dc and ac di erential impedances in optimum gain case. The return loss for device E could not be measured below (2 th )01 (100 MHz for that device), and the I-V curve was not smooth, in contrast to the other devices measured.
Device
L (m)
RN
(
)
dV/dI (
)
RL theory (dB)
RL expt. (dB)
Vdc =Idc
(
)
RL theory (dB)
RL expt. (dB)
0 ! < th1
A1 B D E 0.08 0.16 0.6 3 56 80 93 86 29 35 38 18 12 15 17 7
0 ! < th1
0 ! > th1
29 28 28 14 12 11 11 5
0 ! > th1
>10 >10
8 -
>10 >10 >10
4
Table 5.4: Device dc and ac di erential impedances in overpumped case. The return loss for device E could not be measured below (2 th )01 (100 MHz for that device), and the I-V curve was not smooth, in contrast to the other devices measured.
116
5.3
Normal state noise thermometry measurements
In section 5.1, the crossover from cooling behavior dominated by electron-phonon interaction to cooling behavior dominated by out-di usion was demonstrated by measuring the thermal time constant as a function of device length. However, because of possible complications due to non-equilibrium superconducting and non-uniform heating e ects, it is desirable to investigate this cross-over under more well-understood conditions. Additionally, the crossover from phonon dominated to di usion dominated behavior must be understood if the crossover from dissipative to non-dissipative transport in electronic conductors generally is to be understood. (In addition to the work described in this thesis, the crossover from phonon dominated to di usion dominated to dissipationless transport has been investigated and discussed in de Jong (1995), Kanskar and Wybourne (1994), Prober et al. (1995), Steinbach et al. (1996), Karasik et al. (1996), and Pothier et al. (1997).) The experimental setup described in this thesis is well-optimized to measure the noise generated by the devices very quickly and accurately, as long as the device impedance is not di erent from the device impedance when it is used to do the calibration. The
uctuation-dissipation theorem guarantees that the output noise generated by the device when it is in the normal state and biased at zero voltage is simply kB T per unit bandwidth, as long as it is well-matched to the input impedance of the ampli er. (The match was shown to be very good in the previous section, where the di erential impedance was measured in the normal state.) When the device is in the normal state, an increase in bias voltage does not change the di erential impedance signi cantly, so that the calibrations done under zero bias conditions can be used with impunity. Additionally, since the frequency band is typically 500 MHz, an integration time of only 2 ms can allow a measurement of the output noise with a statistical uncertainty of only 0.1%. Therefore, it is possible to quickly and accurately measure the increase in the electron temperature (averaged over the length of the device) as a function of applied power. The calibration technique described in section 3.5 allows the ampli er contribution to be accurately subtracted o , and the temperature of the device measured using noise-thermometry with an accuracy equal to that of the independent thermometer, which was 50 mK in this case. The output noise and hence electron temperature was measured as a function of applied dc power at a bath temperature above Tc , at 6 or 6.5 K, for several of the devices used in this thesis. The results of these measurements are plotted for four devices in gure 5.48. We consider rst device E, the
117
longest device. Since the length of this device is much longer than
Le0ph , the temperature pro le is
uniform over most of the length of the bridge, except within Le0ph of the ends. We can safely neglect the end-e ect in this case. Plotting the increase in temperature with input power (as is shown in the last graph of gure 5.48) allows determination of the strength and temperature-dependence of the electron-phonon interaction. The power-law of the temperature dependence of the electron-phonon interaction for this device is well-described by equation 2.62, with value is reasonably consistent with the value of
A
= 2:34 1010
W m03 K 04.
This
A
= 0:98 1010
W m03 K 04
found in Gershenzon
et al. (1990) for samples of the same material, thickness, and di usion constant. In the rst graph in gure 5.48, the electron temperature vs. input dc power is plotted for device A1, together with the analytical prediction of equation 2.45 derived in chapter 2 by neglecting the electron-phonon interaction. Since device A1 is suciently shorter than
Le0ph , this analytical
solution describes the data very well. Additionally, a numerical solution to the di usion equation was performed in Chalsani (1997) which included the electron-phonon interaction, with strength given by equation 2.62. The results of this simulation are also plotted.4 The simulation agrees with the analytical solution since the electron-phonon interaction can be safely neglected for this device length. The electron temperature vs. input dc power is plotted for device B in the second graph of gure 5.48, in addition to results of the numerical simulations and the analytical solution neglecting the electron-phonon interaction. At temperatures below
8 K, the electron-phonon interaction
is not signi cant, and the device is still in the di usion-cooled regime. This is evident because the numerical simulation which includes the electron-phonon interaction agrees with the analytical solution which neglects the electron-phonon interaction, and both agree with the data. Above 8 K, the simulations and data deviate from the analytical solution, indicating that the electron-phonon interaction is becoming important. The crossover from di usion dominated to the intermediate behavior where the electron-phonon interaction is signi cant can occur in the same device because the electron-phonon length
Le0ph L
varies with temperature. As was discussed in section 2.3, it is
the ratio of the device length
to
Le0ph
which determines the relative importance of di usion
4 The Lorenz number was varied in Chalsani (1997) for the simulations and analytical solutions plotted in gure 5.48 in order to obtain the agreement shown. The free electron value of L = 2:45 1008 W
K 02 was adjusted to 3:3 1008 W
K 02, which is acceptable since experimentally determined values of L generally depend on temperature and material.
118
and the electron-phonon interaction. The electron temperature vs. input dc power is plotted for device D in the third graph of gure 5.48, as well as results of the numerical simulations and the analytical solution neglecting the electron-phonon interaction. For this device, the analytical solution neglecting the electron-phonon interaction is not valid at any temperature, so that the device is not in the di usion-cooled regime at any temperature. Di usion still contributes signi cantly to the cooling, and it is not until device E that the di usion can be neglected altogether. Thus, the crossover from di usion-cooled to phonon-cooled behavior has been demonstrated in two ways using noise thermometry, in addition to the crossover demonstrated in section 5.1 by measuring the thermal time constant. At a xed temperature of 6 K, varying the device length changes the behavior from phonon to di usion-cooled. For xed device length, varying the temperature changes
Le0ph , which changes the relative importance of di usion and the electron-phonon interac-
tion within the same device in a clearly measurable way. Numerical simulations which include both the e ects of di usion and the electron-phonon interaction agree with all the data by adjusting only two parameters, the Lorenz number and the electron-phonon coupling constant.
119
< Telectron(x) >x (K)
10 Device A1 8 6 0 100
Experimental data Results of num. sim. including e-ph int. Analytical prediction neglecting e-ph int.
200 300 DC Power (nW)
400
500
< Telectron(x) >x (K)
10 8 6 0
Device B
100 Device D
200 300 DC Power (nW)
400
500
< Telectron(x) >x (K)
10 8 6 0
100 Device E
200 300 DC Power (nW)
400
500
< Telectron(x) >x (K)
10 8 6 0
2
4 6 DC Power (µW)
8
10
Figure 5.48: Measured electron temperature vs. dc power using noise thermometry. Note the change in units on the abscissa for device E.
120
5.4
Comparison with theory
coupled
In this section, we compare the measured results of the
output noise and
coupled
conversion
eciency with the theoretical predictions presented in chapter 2. (Later in this section, the di erence between the coupled and available conversion eciency and noise is calculated to be small, typically less than 10%.) There, a simple lumped-element thermal model was developed which neglected the e ect of the non-uniform temperature pro le and non-uniform dissipation of power. This simple model allows a prediction of the output noise and conversion eciency in terms of the electron temperature, the device resistance, the thermal conductance from the electrons to the bath, the dc current, the LO power, and slope of the R vs. T curve, i.e. estimate all of the above parameters except
dR=dTe .
It is possible to measure or
dR=dTe
under the operating conditions.
If the electron temperature could be accurately determined, then the instantaneous value of
dR=dTe
(which depends strongly on the electron temperature near
Tc) could be determined.
How-
ever, there is no clear way to accurately determine the electron temperature, which makes comparison of theory and experiment dicult. Both the conversion eciency and the output noise increase as the square of
dR=dTe , so that an upper limit on both of these parameters can be calculated based dR=dTe .
There are two possible solutions to determining
on the maximum value of
dR=dTe,
and
both depend on the assumption that the I-V curve is determined entirely by the heating process. The physical model is the following. DC (and possibly ac) power heats the electrons above the bath temperature. This causes the resistance to fall somewhere between zero and the value of the electron temperature. Here \resistance" is interpreted as resistance will be di erent from
Rn , depending on
The di erential
Vdc =Idc.
Vdc =Idc
since increasing the current or voltage increases the power
dissipated, which in turn increases the temperature, which in turn increases the resistance, which in turn changes
Vdc =Idc.
A temperature well above
Tc
would cause the resistance to be equal to
Rn . Rn ,
A temperature in the middle of the transition would cause a resistance of approximately 1=2
and so forth. Since the R vs. T curve can be (and was) measured under conditions of negligible self-heating, by measuring the resistance, one can determine the electron temperature, and hence the instantaneous value of
dR=dTe . dR=dTe , as well as the value of dR=dTe
The predicted conversion eciency and output noise based on equations 2.13 and 2.28 was calculated for each device by using the maximum value of
121
determined using the method described above5 . This method was carried out for the dc bias voltage used which minimized the mixer noise in both the overpumped and optimum gain cases. The
parameters for the theoretical calculations are shown in tables 5.5 and 5.6. The results of the calculated conversion eciency based on this method and equations 2.13 and 2.28 are presented in tables 5.7 - 5.8. When the maximum value of
dR=dTe
is used, the conversion eciency is over-
predicted by 6 - 13 dB in all cases. The output noise is also overpredicted by a factor of at least 3-19 for all cases. (The measured noise also includes a component of the Johnson noise.) When the value of
dR=dTe is estimated by inferring the electron temperature from Vdc =Idc , the predicted conversion dR=dTe is used.
eciency and output noise is lower than that predicted when the maximum value of
For the conversion eciency calculated in this way, the predicted conversion eciency is still 7-13 dB higher than that measured, with the exception of device A1 in the optimum gain case, where the agreement is very good. The output noise estimated using the local value of
dR=dTe
is also still
overestimated by a factor of 1.5-8, again with the exception of device A1 in the optimum gain case, where the agreement is very good. Thus, while using the local value of
dR=dTe
to estimate the
output noise and eciency improves the agreement, the theory still overpredicts both quantities by a large factor. There is a second way to estimate the local value of
dR=dTe ,
which also could be expected to
include the e ects of the non-uniform temperature pro le. The method requires only a measurement of the di erential resistance at low frequencies (which can be calculated from the measured I-V curve), and a measurement of the dc current and voltage. An increase in bias voltage increases the power dissipated, hence the electron temperature, hence the resistance. The quantity 0
2 Idc dR=dTe G
can thus be measured. In section 2.6, the prediction for the conversion eciency and output noise was expressed in terms of
0 , the
electron temperature, the thermal conductance, and the dc and
LO power. Therefore, the prediction of the conversion eciency and output noise can be done using a value of
dR=dTe e ectively determined directly from the measured I-V curve. 0
This procedure has
the additional advantage that the dc bias dependence of the gain and noise can also be related to the I-V curve, since can be calculated at each bias point. This procedure has been carried out,
and the resultant theoretical predictions for the gain and noise are compared to the experimental
5 The mismatch factor IF has been calculated assuming an IF ampli er load impedance of 50
. For all devices measured, it varies from 0.87 to 1, with the exception of device E, where it varies from 0.58 to 0.68.
122
results for all ve devices measured in the optimum gain and overpumped cases in gures 5.49 - 5.68. Since the predictions depend on the calculated values of I-V curves for each device in the appendix. For the optimum gain case, the eciency and output noise due to thermal
uctuations are predicted to increase monotonically as the dropback region is approached, which is in agreement with the experimental data. This is intuitively plausible for the following reason: at high bias voltages, the electron temperature is much higher than
and 0 , these are also plotted with the
Tc ,
so that the local value of
dR=dTe Tc ,
is
expected to be low, and hence the conversion eciency and output noise due to thermal
uctuations also. As the bias voltage is reduced, the electron temperature gets closer and closer to where
dR=dTe
is large. There, the conversion eciency and output noise due to thermal
uctuations are
expected to be larger. Therefore, the qualitative agreement is good, but the quantitative agreement is poor. For device A1, the qualitative agreement is not good, either. In the overpumped case, there is also qualitative agreement with the voltage dependence of the conversion eciency and output noise. At zero voltage, the resistance is low but non-zero. This is due to the fact that the LO power has heated the electrons up close to As dc power is applied, the electrons get closer to electrons get close to
Tc,
but not above it.
Tc,
thus increasing the resistance. When the
Tc, the local value of dR=dTe
increases, causing the conversion eciency and
output noise due to thermal
uctuations to increase. As more dc power is applied, the electrons get heated well above
Tc, causing the local value of dR=dTe
to decrease, and hence the conversion
eciency and output noise due to thermal
uctuations to decrease. Thus, there is a voltage at which the conversion eciency and output noise due to thermal
uctuations are maximized, and this is predicted theoretically as well as observed experimentally. In the overpumped case, there is again qualitative agreement between the theory and experiment, but the quantitative agreement is only marginal. The theory presented in chapter 2, to which we have been comparing the experimental data, is a simpli ed phenomenological model of a more complicated physical system. It treats the application of ac and dc power as equivalent, and it treats the system as a lumped-element, neglecting any spatial distribution of energy and temperature. Additionally, the physical mechanism of the resistance vs. temperature curve is not included in the theoretical description. A full theoretical explanation for the
123
broadened transition width has not yet been developed. An understanding of physical mechanisms that cause the nite width of the superconducting transition would need to be developed before a more complete theory of device performance could be achieved. Then, a proper treatment of the problem would need to include the non-equilibrium superconducting dynamics of the system, with the inclusion of the spatial dependence of the local energy distribution function, and the implications of this for the dc and ac voltages developed at the intermediate, rf and LO frequencies, as well as the response to such voltages being applied. The approach taken for the research described in this thesis has been to measure the conversion eciency and output noise, and its dependence on the intermediate frequency, dc and LO powers, and device length. The absolute value of the conversion eciency and noise are not quantitatively predicted by the model presented in chapter 2, where a local temperature is de ned and assumed to determined the resistance. This clearly motivates the need for experimental work to test theoretical assumptions. In contrast, the frequency dependence of the output noise and conversion eciency
do
agree with the theory presented in chapter 2, in addition to the length-dependence of the relevant thermal time constant. The
mixer noise
is actually low, and therefore the di usion-cooled hot-
electron bolometer is an excellent device for use in ultra-sensitive THz receivers.
124
Dev.
Vdc=Idc
(
)
Idc
(A)
Vdc
(mV)
Pdc
(nW)
PLO
(nW)
dR=dTe
(
/K) loc./max.
G3 (nW/K) Exp.(thy.) 40 (29) 30(20) -(17) 44(-)yy 520(-)yyy
y 0
y
z IF
A1 B C D E
22 22 32 24 11
20 15.7 14.1 15.2 60
0.45 0.35 0.46 0.38 0.65
9 5.5 6.4 5.6 39
13 5 8 15 85
68/140 135/200 144/250 -/2505 -/2505
0.16 0.56 0.41 0.66 0.71
0.06 0.22 0.086 0.24 0.46
0.86 0.85 0.95 0.87 0.58
y Determined from measured I-V curve using equations 2.15 and 2.69. z Calculated from equation 2.16. 3 Measured value at 6 K or 6.5 K extrapolated to 5.5 K. (Theoretical value calculated using RN
Table 5.5: Device parameters in optimum gain case.
and equations 2.47.) See section 5.3 for the method of determining G experimentally. 5 Not actually measured. Estimated based on device C, which has the same normal state resistance as devices D,E. yy Expt. value @ 6.5 K was 65 nW/K. 20.5 nW/K was due to di usion cooling and scaled linearly in temperature to 5.5 K. The remaining 44.5 nW/K was due to di usion cooling and scaled as T 3 to 5.5 K. The resultant net thermal conductance at 5.5 K is thus estimated to be 44 nW/K. yyy Expt. value @ 6 K was 675 nW/K. This was scaled as T 3 to 5.5 K in order to estimate the value at Tc , resulting in 520 nW/K.
125
Dev.
Vdc=Idc
(
)
Idc
(A)
Vdc
(mV)
Pdc
(nW)
PLO
(nW)
dR=dTe
(
/K) loc./max.
G3 (nW/K) Exp.(thy.) 40 (29) 30(20) -(17) 48(-)yy 520(-)yyy
y 0
y
z IF
A1 B C D E
28 28 37 28 14
18.1 9.1 10.5 8.2 14.4
0.5 0.25 0.4 0.23 0.2
9 2.3 4.3 1.9 2.9
26 10 16 30 170
103/140 163/200 164/250 -/2505 -/2505
-0.025 0.095 0.15 0.13 0.13
-0.007 0.028 0.021 0.037 0.076
0.92 0.92 0.98 0.92 0.68
y Determined from measured I-V curve using equations 2.15 and 2.69. z Calculated from equation 2.16. 3 Measured value at 6 K or 6.5 K extrapolated to 5.5 K. (Theoretical value calculated using RN
Table 5.6: Device parameters in overpumped case.
and equations 2.47.) See section 5.3 for the method of determining G experimentally. 5 Not actually measured. Estimated based on device C, which has the same normal state resistance as devices D,E. yy Expt. value @ 6.5 K was 65 nW/K. 20.5 nW/K was due to di usion cooling and scaled linearly in temperature to 5.5 K. The remaining 44.5 nW/K was due to di usion cooling and scaled as T 3 to 5.5 K. The resultant net thermal conductance at 5.5 K is thus estimated to be 44 nW/K. yyy Expt. value @ 6 K was 675 nW/K. This was scaled as T 3 to 5.5 K in order to estimate the value at Tc , resulting in 520 nW/K.
126
Device
A13 B C33 D E
(dB) calc. from eq. 2.13 max./local dR/dT used +1.0/-5.3 +0.2/-3.2 +0.7/+0.2 +0.3/+0.3/-
(dB) calc. from eq. 2.70
(dB) expt.
-17.5 -7 -9.4 -0.5 0.0
-5.6 -11 -9.9 -5.4 -8.6
(K) calc. from eq. 2.29 max./local dR/dT used 232/55 384/175 656/218 360/690/-
TT:F:
(K) calc. from eq. 2.71
TT:F:
(K) expt.
Tout
3.5 73 15 174 404
TJohn: )
37 51 44 118 105
(incl.
Table 5.7: Predicted and experimental conversion eciency and output noise in optimum gain case. The measured quantities are for IF=125-215 MHz. Theoretical predictions are for IF=0. This is only a signi cant di erence for devices D,E. Eciencies are SSB. No electro-thermal feedback correction was applied for calculations of eciency and noise using eqs. 2.13 and 2.29, respectively. For devices C and D, the bias voltage used was not exactly where was maximized in order to allow dV/dI to be calculated. 3 I-V was pumped with 18 GHz LO; Tout quoted for 18 GHz LO. was measured relative with 18 GHz LO and normalized to measured value with 20 GHz LO. 33 1.25-1.75 GHz. Spectrum not measured.
Device
A13 B C33 D E
(dB) calc. from eq. 2.13 max./local dR/dT used +2.3/0.0 -2.2/-4.0 +0.7/+0.2 0.0/-7.0/-
(dB) calc. from eq. 2.70
(dB) expt.
-31 -17.2 -13.8 -8.8 -3.7
-7 -13.5 -12.7 -10.4 -20
(K) calc. from eq. 2.29 max./local dR/dT used 160/86 110/73 325/140 87/37/-
TT:F:
(K) calc. from eq. 2.71
TT:F:
(K) expt.
Tout
0.1 3.5 2.6 12 78
TJohn: )
14 14 17 26 10
(incl.
Table 5.8: Predicted and experimental conversion eciency and output noise in overpumped case. The measured quantities are for IF=125-215 MHz. Theoretical predictions are for IF=0. This is only a signi cant di erence for devices D,E. Eciencies are SSB. No electro-thermal feedback correction was applied for calculations of eciency and noise using eqs. 2.13 and 2.29, respectively. 3 I-V was pumped with 18 GHz LO; Tout quoted for 18 GHz LO. was measured relative with 18 GHz LO and normalized to measured value with 20 GHz LO. 33 1.25-1.75 GHz. Spectrum not measured.
127
4
Conversion efficiency (SSB)
2
0.18
6 4 2
Experimental data IF = 1.25-1.75 GHz
0.018
6 4 2
Theoretical prediction IF = 0
0.001
-1.2
-0.8
-0.4 0.0 0.4 Voltage (mV)
0.8
1.2
Figure 5.49: Comparison of theoretical and experimental eciency for device A1 in optimum gain case.
60 50
Measured total output noise IF = 1.25-1.75 GHz
Noise tempeature (K)
40 30 20 10 0 -1.2 -0.8 -0.4 0.0 0.4 Voltage (mV) 0.8 1.2
Predicted thermal fluctuation noise component of output noise IF = 0
Figure 5.50: Comparison of theoretical and experimental output noise for device A1 in optimum gain case.
128
Conversion efficiency (SSB)
0.1
Experimental data IF = 1.25-1.75 GHz
0.01
0.001
Theoretical prediction IF = 0
0.0001
-1.2
-0.8
-0.4 0.0 0.4 Voltage (mV)
0.8
1.2
Figure 5.51: Comparison of theoretical and experimental eciency for device A1 in overpumped case.
30 25
Measured total output noise IF = 1.25-1.75 GHz
Noise tempeature (K)
20 15 10 5 0 -1.2 -0.8 -0.4 0.0 0.4 Voltage (mV) 0.8 1.2
Predicted thermal fluctuation noise component of output noise IF = 0
Figure 5.52: Comparison of theoretical and experimental output noise for device A1 in overpumped case.
129
1
Conversion efficiency (SSB)
6 4 2
0.1
6 4 2
Theoretical prediction IF = 0
0.01
6 4 2
Experimental data IF = 125-215 MHz
0.001 -2.0
-1.5
-1.0
-0.5 0.0 0.5 Voltage (mV)
1.0
1.5
2.0
Figure 5.53: Comparison of theoretical and experimental eciency for device B in optimum gain case.
140
Noise tempeature (K)
120 100 80 60 40 20 0 -2.0 -1.5 -1.0
Predicted thermal fluctuation noise component of output noise IF = 0
Measured total output noise IF = 125-215 MHz
-0.5 0.0 0.5 Voltage (mV)
1.0
1.5
2.0
Figure 5.54: Comparison of theoretical and experimental output noise for device B in optimum gain case.
130
0.1
Conversion efficiency (SSB)
8 6 4 2
Theoretical prediction IF = 0
0.01
8 6 4 2
Experimental data IF = 125-215 MHz
0.001
-2
-1
0 Voltage (mV)
1
2
Figure 5.55: Comparison of theoretical and experimental eciency for device B in overpumped case.
25 20
Predicted thermal fluctuation noise component of output noise IF = 0 Measured total output noise IF = 125-215 MHz
Output noise (K)
15 10 5 0 -2
-1
0 Voltage (mV)
1
2
Figure 5.56: Comparison of theoretical and experimental output noise for device B in overpumped case.
131
Conversion efficiency (SSB)
Theoretical prediction IF = 0
0.1
Experimental data IF = 1.25-1.75 GHz
0.01
0.001
0.0001 -2.0
-1.5
-1.0
-0.5 0.0 0.5 Voltage (mV)
1.0
1.5
2.0
Figure 5.57: Comparison of theoretical and experimental eciency for device C in optimum gain case.
60 50
Measured total output noise IF = 1.25-1.75 GHz Predicted thermal fluctuation noise component of output noise IF = 0
Noise tempeature (K)
40 30 20 10 0 -2.0
-1.5
-1.0
-0.5 0.0 0.5 Voltage (mV)
1.0
1.5
2.0
Figure 5.58: Comparison of theoretical and experimental output noise for device C in optimum gain case.
132
4
Conversion efficiency (SSB)
2
0.18
6 4 2
Theoretical prediction IF = 0
Measured efficiency (SSB) IF=1.25-1.75 GHz
0.018
6 4 2
0.001 -2.0
-1.5
-1.0
-0.5 0.0 0.5 Voltage (mV)
1.0
1.5
2.0
Figure 5.59: Comparison of theoretical and experimental eciency for device C in overpumped case.
25 20 15 10 5 0 -2.0
Measured total output noise IF = 1.25-1.75 GHz Predicted thermal fluctuation noise component of output noise IF = 0
Noise tempeature (K)
-1.5
-1.0
-0.5 0.0 0.5 Voltage (mV)
1.0
1.5
2.0
Figure 5.60: Comparison of theoretical and experimental output noise for device C in overpumped case.
133
1
Conversion efficiency (SSB)
6 4 2
Measured efficiency IF=125-215 MHz
0.1
6 4 2
Theoretical prediction IF = 0
0.01
6 4 2
0.001 -2.0
-1.5
-1.0
-0.5 0.0 0.5 Voltage (mV)
1.0
1.5
2.0
Figure 5.61: Comparison of theoretical and experimental eciency for device D in optimum gain case.
200
Noise tempeature (K)
150
Measured total output noise IF = 125-215 MHz
100
Predicted thermal fluctuation noise component of output noise IF = 0
50
0 -2.0
-1.5
-1.0
-0.5 0.0 0.5 Voltage (mV)
1.0
1.5
2.0
Figure 5.62: Comparison of theoretical and experimental output noise for device D in optimum gain case.
134
1
Conversion efficiency (SSB)
6 4 2
Theoretical prediction IF = 0
0.1
6 4 2
Measured efficiency IF = 125-215 MHz
0.01
6 4 2
0.001 -2.0
-1.5
-1.0
-0.5 0.0 0.5 Voltage (mV)
1.0
1.5
2.0
Figure 5.63: Comparison of theoretical and experimental eciency for device D in overpumped case.
100 80
Output noise (K)
60 40 20 0 -2.0
Predicted thermal fluctuation noise component of output noise IF=0
Measured total output noise IF = 125-215 MHz
-1.5
-1.0
-0.5 0.0 0.5 Voltage (mV)
1.0
1.5
2.0
Figure 5.64: Comparison of theoretical and experimental output noise for device D in overpumped case.
135
1
Conversion efficiency (SSB)
6 4 2
0.1
6 4 2
Measured efficiency IF = 125-215 MHz
0.01
6 4 2
Theoretical prediction @ IF = 0
0.001
-8
-6
-4
-2 0 2 Voltage (mV)
4
6
8
Figure 5.65: Comparison of theoretical and experimental eciency for device E in optimum gain case.
140
Noise tempeature (K)
120 100 80 60 40 20 0 -8 -6
Predicted thermal fluctuation noise component of output noise Measured total output noise IF = 0 IF = 125-215 MHz
-4
-2 0 2 Voltage (mV)
4
6
8
Figure 5.66: Comparison of theoretical and experimental output noise for device E in optimum gain case.
136
4
Conversion efficiency (SSB)
Theoretical prediction IF = 0
2
0.18
6 4 2
0.018
6 4 2
Measured efficiency IF = 125-215 MHz
0.001
-8
-6
-4
-2 0 2 Voltage (mV)
4
6
8
Figure 5.67: Comparison of theoretical and experimental eciency for device E in overpumped case.
100 80
Predicted thermal fluctuation noise component of output noise IF = 0 Measured total output noise IF = 125-215 MHz
Output noise (K)
60 40 20 0 -8
-6
-4
-2 0 2 Voltage (mV)
4
6
8
Figure 5.68: Comparison of theoretical and experimental output noise for device E in overpumped case.
Chapter 6
Comparison to JPL THz Receiver Measurements
The motivation for the work described in this thesis is the ultimate development of an ultra-low noise THz receiver. The experiments performed with 20 GHz signals were designed to investigate the physical processes which determine the behavior of the device. In this chapter, we compare the results presented in this thesis with results on similar devices measured in the research group of W.R. McGrath at JPL with 500 GHz (Skalare et al. 1996), 1.2 THz (Skalare et al. 1997), and 2.5 THz (Karasik et al. 1997) signals in order to determine the relevance of the measurements presented in this thesis to actual THz receivers. From rst principles, there are three main frequency scales in the problem which may be important. The rst frequency scale is the inverse of the thermal time constant, and this issue has already been extensively discussed. The second frequency scale is the gap frequency, which is approximately 740 GHz for bulk Nb. The most important issue regarding the gap frequency scale is whether the device responds to ac power below the gap frequency in the same way as it does to ac power above the gap frequency, i.e. is the conversion eciency measured in this thesis with 20 GHz signals indicative of what can be achieved with THz signals? An additional frequency scale is the inverse of the momentum-relaxation rate, which is approximately 100 THz for the dirty lms measured in this thesis. If the ac voltage changes faster than this frequency, an inductive component of the device impedance develops, which decreases the absorption of rf power, 137
138
and hence the conversion eciency. Experiments done by Gershenzon et al. (1982) indicate that the absorption of power is constant in thin- lm Nb up to approximately 100 THz, and this frequency scale is not approached by the experiments done at Yale and JPL. If a voltage is applied to the device with a frequency above the gap frequency, then the power dissipated in the device is uniform along the length of the device, regardless of whether it is in the superconducting, normal, or intermediate state. (This is because an rf photon has enough energy to break a Cooper pair, which results in the absorption of rf power.) In contrast, voltages applied below the gap frequency cause dissipation of power only where the electrons are in the normal or intermediate state, which is generally not spatially uniform for the devices studied in this thesis. For the thin- lm Nb used in the Yale and JPL experiments, the gap frequency is lower then the bulk value since
Tc is lower.
For a
Tc of 5.5 K, the gap frequency is approximately 440 GHz.
Near
Tc the
energy gap is suppressed, and for bulk superconductors the energy gap (and hence the frequency scale associated with it) approaches zero as the temperature approaches Tc . Since the devices studied here are operated very near
Tc
(which is not well-de ned due to the nite-transition width), it is
possible that the dissipation of power is uniform, in which case there should be no di erence between the response to 10 GHz signals and THz signals. Since there is no clear theoretical prediction, the issue must be investigated experimentally. Experiments performed at JPL were done on thin lm Nb devices which were fabricated using the same process as the devices studied in this thesis.1 In the 500 GHz JPL experiments, the rf coupling to the device was accomplished with a waveguide with two adjustable tuning elements, which allow for variation of the rf and LO source impedance to optimize the rf coupling from the receiver input to the device. The 1200 GHz and 2.5 THz experiments were done quasi-optically with antennas integrated onto the substrate, so that the rf coupling to the device could not be adjusted. In all the JPL experiments, the rf coupling eciency to the device was estimated in order to deconvolute the rf coupling loss from the intrinsic device conversion eciency. The output noise was also measured in the JPL experiments over the 1.25-1.75 GHz frequency range. A summary of the conversion eciency and output noise measured at Yale and JPL is presented in table 6. The device parameters are also presented. The devices measured at JPL included normal
1 The devices used in the 500 GHz experiments performed at JPL were not fabricated using the self-aligned process described in chapter 3, but with a similar process.
139
state resistances that were both larger and smaller than those measured at Yale, and the device lengths of the JPL devices was similar to those measured at Yale. The bias conditions for the Yale results presented are those of optimum gain. The data from the JPL experiments was not easily classi ed into the \optimum gain" and \overpumped" cases, and further experiments will need to be done to clarify if a similar classi cation scheme is valid for the response to THz signals. Therefore, the data presented in table 6 was for the bias conditions which minimized the system or receiver noise performance. These conditions resemble the \optimum gain" case for the 500 GHz experiments, in that the I-V curve was hysteretic. However, for the 1200 GHz and 2.5 THz experiments, the I-V curve did not clearly resemble the I-V curves presented in this thesis. The measured conversion eciency at Yale was between -6 dB and -11 dB, while the JPL results nd between -14 dB and -18 dB. This suggests that the measured conversion eciency at 20 GHz may be larger than that to be expected at THz frequencies, by 7-8 dB. The output noise measured at JPL varied from 10-41 K, while the Yale results found a between 49 K and 55 K. It is dicult to draw conclusions from these data, since the output noise depends on the dc and lo power applied in all experiments. For this reason, the output noise measured in the absence of any applied ac power is also indicated in table 6 for those cases in which the data are available. Here, in both cases, the output noise is maximum when the dc bias is very close to the point of instability, as in the \optimum gain" case. The JPL data fall between 12 K and 37 K, while the Yale data fall between 30 K and 57 K. Thus, the JPL data are somewhat smaller. This may be due to the fact that the devices measured at JPL had a slightly di erent R vs. T characteristic since they were fabricated from di erent lms than the devices measured at Yale. Since the measured output noise is di erent when no LO power is applied, it should not be expected to be the same when LO power is applied. The output noise, then, is similar but not exactly the same, and the conversion eciencies measured at JPL are also smaller. The estimated mixer noise for the Yale experiments is between 120 K and 320 K (DSB), while it is between 210 K and 560 K for the JPL experiments. Thus, it is possible that the value of dR/dT for the JPL devices was smaller, causing a smaller eciency and output noise, but a similar mixer noise. The gain bandwidth has been carefully measured at JPL in the 500 GHz experiments, and it is found to be 1.9 GHz. This point is included in the plot of bandwidth vs. length in gure 5.26.
140
Frequency (GHz)
E . (dB,SSB)
Tout
(K)
Tmix
(K,DSB)
Tout
(K) (no LO pwr.)
L (m)
RN
(
)
Rsheet
(
)
Gain BW (GHz)
20? 20? 20? 533 1267 2500
-5.6 -11 -8 -14.4 35
49 57 55 415 16.6y 10z
120 320 200 5605 166y 350z
57 30.6 36.7 13.7
0.08 0.16 0.24 0.27 0.3 0.3
56 80 96 20 140 23
29 29 29 10.4 70 11.5
>6
2.4 1.5 1.7 -
6 -1363y -18.563z
< 12
Table 6.1: Comparison of JPL and Yale mixer results. The upper half of the table represents the results presented in this thesis measured at Yale, while the lower half represents the JPL data (Skalare 1997; Karasik 1997). The contribution of the IF ampli er has been subtracted o . ? E . referred to device. Bias conditions for optimum gain used. Output noise is extrapolated to f = 0. Sheet resistance determined from larger device on wafer. 5 Referred to dewar input, incl. 1 dB cplg. loss. Output noise is at 1.24-1.56 GHz. y Referred to device, does not incl. 9.1 dB cplg. loss. Output noise is at 1.24-1.56 GHz. z Referred to device, does not incl. 8.5 dB cplg. loss. Output noise is at 1.5 GHz. The 2.5 THz measurements were done at a bath temperature of 4.2 K, in contrast to the other data in the table where the bath temperature was approximately 2 K. There, the JPL measurement is in good agreement with the data presented in this thesis. Since the sheet resistance for the JPL device is approximately 3 times smaller for the JPL device than the Yale devices, the agreement should not be as good as it is. However, errors in the estimate of the sheet resistance due to errors in the estimate of the lm thickness could account for some of the discrepancy. It is possible that the electro-thermal slowing factor is di erent in the JPL and Yale experiments, but this would not account for a factor of 3 discrepancy. This may indicate that more complicated non-equilibrium processes conspire to enforce the scaling law that we nd experimentally in this thesis, regardless of the sheet resistance. In gure 6.1, the system noise temperature is plotted as a function of rf frequency for several di erent detector technologies. From this graph, it is clear that the hot-electron bolometers are already the lowest-noise mixers available for rf frequencies above approximately 1 THz.
141
105
Receiver Noise Temp. (K,DSB)
104
SIS Nb HEB NbN HEB Schottky, 300 K Schottky, 20 K InSb HEB
103
102
Quantum limit, hυ/k
10
1
10
2
2
3
4
5 6 7 89
10 Frequency (GHz)
3
2
3
4
5 6 7 89
104
(Adopted
Figure 6.1: Receiver noise temperature vs. rf frequency for various technologies. from Schoelkopf et al. (1997) and Karasik et al. (1997).)
Chapter 7
Conclusions
7.1 Summary of results presented in this thesis
In this thesis, the physical processes that govern the performance of di usion-cooled hot-electron bolometers have been systematically investigated in a series of experiments measuring the conversion eciency, output noise, and device di erential impedance under a variety of experimental conditions for devices of several di erent lengths. In chapter 2, a simple theoretical model was presented which treats the measured resistance vs. temperature curve phenomenologically and predicts the output noise, conversion eciency, and device di erential impedance as a function of the dc and ac power, the thermal conductance of the electrons to the bath, the electron temperature, and the normal state resistance. An experimental setup was designed and constructed to allow the measurements of the conversion eciency and output noise of a suite of devices with LO and rf frequencies from 8 to 40 GHz. A broadband coupling scheme to convert from coax to microstrip and then to the device through a \
ip-chip" method was designed and constructed, allowing resonant free coupling to the devices from dc to 40 GHz. A cryogenically cooled ampli er with input noise of 25 K from 0.1-8 GHz was used to measure coherent signals generated by the device at the intermediate frequency as well as incoherent noise generated by the device. Two calibration schemes were devised and shown to agree. The
in-situ
calibration scheme used the Johnson noise from the device when it was biased above 142
Tc
143
as a source of known absolute power to calibrate the ampli er gain and noise. In this way, the noise generated by the device could be measured by subtracting o the ampli er noise contribution. The conversion eciency and output noise were measured with a bath temperature of 2 K as a function of dc bias voltage for a variety of devices in two cases: when enough LO power was applied to completely suppress the critical current (the overpumped case), and when the LO power was adjusted to maximize the (coupled) conversion eciency. The
mixer noise
in the two cases was
very similar, but the output noise and conversion eciency were higher by a factor of 2-3 in the optimum gain case than the overpumped case. The conversion eciencies measured ranged from -6 dB to -11 dB in the optimum gain case, and the output noise ranged from 50 K to 230 K. In the overpumped case, the conversion eciencies ranged from -10 dB to -13.5 dB, while the output noise ranged from 13-70 K. These results did not depend sensitively on the LO frequency used as long as it was larger than (2 th )01 . A dramatic enhancement of the noise was found for device A1 in the overpumped case when the applied LO frequency was below (2 th )01 . The measured output noise and conversion eciency were compared with the theoretical predictions of chapter 2 in two ways: rst, the value of
dR=dTe
was estimated from the measured R
vs. T curve, and the output noise and conversion eciency was overpredicted in all cases using this method. A second method used the measured dc I-V curve to estimate
dR=dTe ,
and this method
overpredicted the output noise in some cases and underpredicted it in others. However, the qualitative dependence on bias voltage was well predicted, and this is in accord with the general principles of the device operation. The mixer noise was calculated by dividing the measured output noise by the measured conversion eciency, and found to be between 120 K and 530 K. This low noise suggests that the di usion-cooled hot-electron bolometers are the best candidates for ultra-sensitive THz receivers. The dependence of the gain and output noise on intermediate frequency was also measured for the suite of devices. The conversion eciency and output noise decrease when the intermediate frequency is larger than the inverse of the thermal time constant. The mechanism which determines the thermal time-constant depends on the device length. For devices longer than the electron-phonon length
Le0ph , the thermal time-constant is given by the electron-phonon interaction rate.
This time
constant allows for a gain bandwidth of approximately 100 MHz in Nb, which is too slow for practical
144
applications. If the device length is less than
Le0ph ,
then out-di usion of heat into the highly
conducting normal metal leads can dominate the cooling process. In this case, the thermal time constant is given by the di usion time, and scales as the device length squared. The time constants were determined for devices of di erent lengths by tting the conversion eciency as a function of intermediate frequency. The scaling with device length obeyed the theoretical predictions, and the crossover from phonon dominated to di usion dominated behavior was observed as the length of the device varied. A factor of 50 improvement in the gain bandwidth was achieved by using di usion instead of phonon cooling. The shortest device had a gain bandwidth of at least 6 GHz, which is the largest bandwidth achieved in a low-Tc hot-electron mixer as of the writing of this thesis. Since the component of the output noise due to thermal
uctuations was found to decrease with intermediate frequency, the
mixer noise
did not begin to increase until the thermal
uctuation noise component
was smaller than the Johnson noise component. This implies that the noise bandwidth was larger than the gain bandwidth by a factor of 1.4-9.4, which may be useful in actual receivers. Thus, while the absolute conversion eciency and output noise are not well-predicted by the theory, the dependence on intermediate frequency is. The crossover from phonon to di usion dominated behavior was also observed by using noise thermometry in the normal state to measure the increase in electron temperature as a function of applied dc power. The crossover was observed in a single device by varying the device temperature, which changes Le0ph and hence the ratio L=Le0ph . The ratio L=Le0ph determines whether di usion or phonon cooling dominates. The crossover was also observed by measuring the response to dc power of devices of varying lengths at approximately the same temperature of 6 K. Additionally, the device di erential impedance was measured, using a scalar measurement technique. The rf coupling to the devices was measured to be better than approximately 90% for all devices both above and below (2 th )01 , with the exception of device E, where the coupling was only approximately 50% at frequencies below (2 th )01 . These measurements are in agreement with the theoretical prediction that the impedance above (2 th )01 is below (2 th )01 is simply
Vdc =Idc,
and that the impedance
dV=dI .
Since the ultimate goal for di usion-cooled hot-electron bolometers is operation in ultra-sensitive THz receivers, the data presented in this thesis was compared to the data measured on similar
145
devices at JPL with rf frequencies of 500 GHz, 1.2 THz, and 2.5 THz. The JPL measurements nd device conversion eciencies between -13 and -18 dB, which is approximately 8 dB lower than the measurements for this thesis work. The output noise measured at JPL was also lower, varying from 12 K to 37 K. One possible reason for the discrepancy could be a lower value of
noise
dR=dTe .
The
mixer
was similar to that found here, ranging from 210 K to 560 K (DSB). Additionally, the measured
gain bandwidth of 1.9 GHz for a 0.27
m
device is in good agreement with the devices measured
at Yale of similar length, even though the di usion constant of the JPL devices is estimated to be di erent by a factor of approximately 3. The device performance measured at JPL at 1.2 THz and 2.5 THz already indicates that the devices are the best available mixers above 1 THz.
7.2
Suggestions for future experiments
In this thesis, we have investigated the dependence of the gain bandwidth and noise bandwidth on device length, for a xed di usion constant and with negligible electro-thermal feedback. The dependence of the thermal time constant on the di usion constant for devices of xed length with di erent di usion constants would be a complimentary set of experiments to those presented in this thesis. Additionally, by adjusting the width and keeping the length xed, the device resistance and hence electro-thermal feedback factor could be changed, without changing the \bare" time constant. It may be possible to engineer electro-thermal feedback e ects to improve the conversion eciency, but the output noise is also predicted to increase in this case. However, this would reduce the contribution of the IF ampli er noise to the system, and so may be worth investigating. The coupling of the device at the rf, LO, and IF frequencies to a 50
system also would change with device resistance, and this e ect (which is comparable in magnitude to the electro-thermal feedback e ect) would need to be corrected for, making the experiments somewhat challenging. The noise performance of the devices measured for this thesis and at JPL is already excellent for THz receivers. However, it is still at least an order of magnitude above the quantum limit. According to equation 2.31, the mixer noise should scale as with
GTe2 =PLO .
For di usion-cooled devices, G scales
Te , and PLO scales approximately inversely with Te , so that lower Tc devices should have lower Tc materials.
mixer noise. Therefore, the noise performance should improve by using lower
Finally, the temperature pro le is a dicult quantity to measure experimentally, but an impor-
146
tant issue in di usion-cooled bolometers. If lower
Tc
devices are used where the relevant length
scales are longer than those for Nb, such as the electron-electron and electron-phonon interaction lengths, it might be possible to probe the electron energy distribution function along the length of the bridge with a series of normal and/or superconducting tunnel junctions. This would allow the non-equilibrium processes to be investigated as they develop along the length of the bridge.
Appendix A
Calculated values of and 0 from I-V curve.
147
148
1.0 40 20 0.6 0 -20 -40 -1.2 -0.8
Figure A.1:
Current
0.8
Current (µA)
α
α0
0.4 0.2 0.0
α
-0.4 0.0 0.4 Voltage (mV)
0.8
1.2
for device A1 in optimum gain case.
40
Current
0.8
α0
Current (µA)
20 0
0.4 0.0
α
α
-20 -40 -1.2 -0.8
Figure A.2:
-0.4 -0.8 -0.4 0.0 0.4 Voltage (mV) 0.8 1.2
for device A1 in overpumped case.
149
1.0 80 0.8 40
Current
Current (µA)
0.6
α0
0 -40
α
α
0.4 0.2 0.0
-80 -2.0 -1.5 -1.0 -0.5 0.0 0.5 Voltage (mV) 1.0 1.5 2.0
Figure A.3:
for device B in optimum gain case.
40 20 0 -20 -40 -2 -1
α0
0.8 0.4 0.0
α
Current (µA)
α
-0.4 -0.8
Current
0 Voltage (mV)
1
2
Figure A.4:
for device B in overpumped case.
150
80 40 0 -40
Current
α0
0.8 0.4 0.0
α
Current (µA)
α
-0.4 -0.8
-80 -2.0 -1.5 -1.0 -0.5 0.0 0.5 Voltage (mV) 1.0 1.5
2.0
Figure A.5:
for device C in optimum gain case.
40 20 0 -20
α0
0.8 0.4 0.0
α
Current (µA)
α
-0.4 -0.8
Current
-40 -2.0 -1.5 -1.0
Figure A.6:
-0.5 0.0 0.5 Voltage (mV)
1.0
1.5
2.0
for device C in overpumped case.
151
300 0.8 200
α0
Current (µA)
100 0 -100 -200
Current α
0.4 0.0 -0.4 -0.8
α
-300 -2.0
-1.5
-1.0
-0.5 0.0 0.5 Voltage (mV)
1.0
1.5
2.0
Figure A.7:
for device D in optimum gain case.
30 0.8 20
α0
Current (µA)
10 0 -10
Current α
0.4 0.0 -0.4 -0.8
α
-20 -30 -2.0 -1.5 -1.0
Figure A.8:
-0.5 0.0 0.5 Voltage (mV)
1.0
1.5
2.0
for device D in overpumped case.
152
200 150 100
α0 Current
0.8 0.4 0.0
Current (µA)
50 0 -50 -100 -150 -200 -8 -6 -4
Figure A.9:
α
α
-0.4 -0.8 -2 0 2 Voltage (mV) 4 6 8
for device E in optimum gain case.
80 40 0
Current α0
0.8 0.4 0.0
Current (µA)
α
α
-40 -80 -8 -6 -4
Figure A.10:
-0.4 -0.8 -2 0 2 Voltage (mV) 4 6 8
for device E in overpumped case.
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